Generating minimal redundant and maximal irredundant sets in incidence graphs

Generating minimal redundan t and maximal irredundan t sets in incidence graphs ∗ Eman uel Castelo 1 , Jérémie Chalopin 1 , Oscar Defrain 1 , and Simon Vilmin 1 1 Aix-Marseille Univ ersité, CNRS, LIS, Marseille, F rance. F ebruary 23, 2026 Abstract It has b een pro ved b y Boros and Makino that there is no output-p olynomial- time algorithm en umerating the minimal redundant sets or the maximal irredundan t sets of a hypergraph, unless P = NP . The same question was left open for graphs, with only a few tractable cases known to date. In this pap er, w e fo cus on graph classes that capture incidence relations such as bipartite, co-bipartite, and split graphs. Concerning maximal irredundant sets, we show that the problem on co-bipartite graphs is as hard as in general graphs and tractable in split and strongly orderable graphs, the latter b eing a generalization of c hordal bipartite graphs. As formeta minimal redundan t sets enumeration, we ﬁrst show that the problem is in tractable in split and co-bipartite graphs, answ ering the aforementioned op en question, and that it is tractable on ( C 3 , C 5 , C 6 , C 8 ) -free graphs, a class of graphs incomparable to strongly orderable graphs, and which also generalizes chordal bipartite graphs. Keyw ords: enumeration algorithms, maximal irredundant sets, minimal redundan t sets, incidence graphs 1 In tro duction Irr e dundant sets in hypergraphs are those sets of vertices I for which every elemen t x ∈ I has a priv ate edge, i.e., an edge whose intersection with I is exactly { x } . Equiv alen tly , they are often presented as sets of vertices that are minimal with resp ect to the set of h yp eredges they in tersect. Sets of vertices that are not irredundant are called r e dundant . In graphs, irr e dundant sets are deﬁned as the sets of vertices having priv ate neighbors, that is, neigh b ors that are not adjacent to other vertices in the set, allo wing a vertex to b e its own priv ate neighbor. R e dundant sets are deﬁned analogously as the subsets of v ertices that are not irredundant. Redundan t and irredundan t sets of a graph should not b e confused with the redundant and irredundan t sets of the underlying h yp ergraph where eac h edge deﬁnes a h yp eredge. Rather, they correspond to the redundant and irredundan t ∗ The ﬁrst, third, and last authors ha ve b een supp orted b y the ANR pro ject P ARADUAL (ANR-24- CE48-0610-01). 1 sets of the hypergraph of closed-neighborho o ds of the graph, in an analogue of what is domination to transversalit y; we refer to Section 2 for a more in-depth presentation of these notions together with some examples. Irredundancy in graphs was ﬁrst introduced by Co ck ayne, Hedetniemi, and Miller in [ CHM78 ] for its links with indep endence and domination, where it was in particular noted that every maximal indep enden t set is a minimal dominating set, which in turn is a maximal irredundant set. Due to its relation to indep endence and domination, irredundancy in graphs has b een extensively studied [ BC79 , CFPT81 , BC84 , F av86 , BG88 , GL93 , JP90 ]. Let us mention that computing the minim um or maximum size of a maximal irredundan t set is known to b e NP -complete [LP83, HLP85, FFHJ94]. The generalization of irredundancy to hypergraphs has b een considered for algorithmic purp oses in the contexts of minimal domination or transversalit y [ KLM + 15 , BDMI24 ], for it is known that minimal dominating sets (resp. minimal tranv ersals) are those dominating sets (resp. tran versals) which are irredundant. Despite the interest for irredundant sets, little is known regarding their enumeration and that of their dual, redundant sets. Here the goal is, given a graph or a hypergraph, to list without rep etition all its maximal irredundan t sets or minimal redundant sets. In the follo wing, let us denote by GraphMirr · Enum , GraphMred · Enum , HypMirr · Enum , and HypMred · Enum these problems (see Section 2 for the formal deﬁnition). Since the num b er of minimal redundant or maximal irredundant sets can b e exp onential in the size of the input (hyper)graph, ev aluating the complexity of an enumeration algorithm solving either of these problems using the input size only is not a relev ant eﬃciency measure. Instead algorithms are analyzed using output-sensitive c omplexity that estimates the running time of an algorithm in terms of b oth its input and output sizes. An algorithm runs in output-p olynomial time if its execution time is p olynomially b ounded b y the combined sizes of the input and the output. This notion can b e further detailed to analyze the delay b etw een tw o solutions. Namely , we say that an algorithm runs in incr emental-p olynomial time if it outputs the i -th solution after a time b eing p olynomial in the input size plus i . If the dela y b etw een b efore the ﬁrst output, b etw een t wo consecutive outputs, and after the last output is p olynomial in the input size only , then the algorithm is said to run with p olynomial delay . W e redirect the reader to [ Str19 ] for further details on en umeration complexity . W e now review the state of the art concerning the four ab o ve-men tioned problems. The complexity status of HypMirr · Enum and HypMred · Enum was ﬁrst stated as an op en problem b y Uno during the Loren tz W orkshop on “En umeration Algorithms using Structure” held in 2015 [ BBHK15 ]. Shortly after, Boros and Makino announced the follo wing as a negative answer for b oth problems [ BM16 , BM24 ], where the dimension of a h yp ergraph is the maximum size of its edges. Theorem 1.1 (Reformulation of [ BM24 , Theorems 1 & 2]) . Ther e is no output-p olynomial- time algorithm for HypMirr · Enum and HypMred · Enum unless P  = NP , even when r estricte d to hyp er gr aphs of dimension at most thr e e. Let us p oint that the original formulations in [ BBHK15 , BM16 , BM24 ] are stated in the context of redundant and irredundant subh yp ergraphs, 1 whic h is equiv alent to ours up 1 A set of hyperedges (or subhypergraph) is called irre dundant if each of its hyperedge contains a priv ate vertex, i.e., a vertex that is only contained in this h yp eredge within the set; it is called r edundant otherwise. 2 to considering the transp osed 2 h yp ergraph. The ab ov e statement is thus a reform ulation to our context of vertex subsets, as will be the follo wing summary of their result. In addition to Theorem 1.1, the authors in [ BM16 , BM24 ] show that HypMred · Enum and HypMirr · Enum resp ectively admit p olynomial and output-p olynomial times algorithms in h yp ergraphs of bounded de gr e e which is the n umber of edges a vertex intersects. When restricted to hypergraphs of dimension at most tw o, they sho w that HypMred · Enum can b e solved in p olynomial time, while HypMirr · Enum is equiv alent to h yp ergraph dualization, arguably one of the most imp ortant op en problem in enumeration for which no b etter than incremental-quasi-polynomial time is known [ EG95 , FK96 , EMG08 ]. Let us stress again the fact that the results for h yp ergraphs of dimension at most tw o do not apply to GraphMred · Enum nor GraphMirr · Enum , as (ir)redundancy in graphs corresp ond to (ir)redundancy in the h yp ergraph of closed-neigh b orho o ds of these graphs, for whic h the dimension is not b ounded. Later in [ CKMU19 ], the authors study HypMirr · Enum under the prism of parame- terized complexity and in particular derive a p olynomial-delay algorithm on hypergraphs (hence graphs) of b ounded degeneracy . In [ BCO21 ], it is sho wn that GraphMirr · Enum admits an algorithm running with linear delay after p olynomial-time prepro cessing and p olynomial space when restricted to circular-p erm utation and p ermutation graphs. Finally , let us brieﬂy mention that irredundant sets enumeration has triggered some in terest in the con text of input-sensitiv e enumeration [ BRBC + 11 , GKLS19 , GKS19 ], where the goal is to devise exp onential time algorithms minimizing the base of the exp onen t. In this work, we contin ue the research direction that was initiated in [ BCO21 ], as a restriction of the original question ask ed in [ BBHK15 ], and wonder whether GraphMred · Enum and GraphMirr · Enum are tractable. Namely , we wish to answer the following question. Question 1.2. Can GraphMred · Enum and GraphMirr · Enum b e solve d in output- p olynomial time on gener al gr aphs? W e note that the same question is p osed as an op en direction in [CKMU19]. In the light of Theorem 1.1, and driven by the in tuition that the restrictions of HypMred · Enum and HypMirr · Enum to graphs should still b e intractable, 3 w e address Question 1.2 fo cusing on graph classes that capture incidence relations of hypergraphs suc h as bipartite, co-bipartite, and split graphs; we refer to Section 2 for their relation to h yp ergraphs. First, we focus on maximal irredundant sets. On split graphs, it w as already claimed by Uno in [ BBHK15 ] that maximal irredundant sets are in bijection with minimal dominating sets. W e include a pro of of this statemen t for completeness. W e thus derive the following using the algorithm from [ KLMN14 ] for minimal dominating sets enumeration in this graph class. 2 The tr ansp ose d hyper gr aph is obtained by considering each edge as a v ertex, and each v ertex as an edge, while keeping their mutual incidences. In terms of its incidence bipartite graph, it amounts to “swap” its t wo sides. In particular, the dimension of the h yp ergraph b ecomes the degree of its transp osed hypergraph, and its degree becomes its dimension. 3 In what w ould be an analogue of transversals to dominating sets [KLMN14]. 3 Theorem 1.3 ([ BBHK15 ]) . Ther e is a bije ction b etwe en maximal irr e dundant sets and minimal dominating sets in split gr aphs. Conse quently, GraphMirr · Enum c an b e solve d with line ar delay and sp ac e on this class. As for co-bipartite graphs, we obtain the follo wing negative result, which shows the problem to b e c hallenging in this class. Theorem 1.4. Ther e is an output-p olynomial-time algorithm solving GraphMirr · Enum on gener al gr aphs if and only if ther e is one solving the same pr oblem on c o-bip artite gr aphs. T o obtain Theorem 1.4 we consider the hypergraph of closed neigh b orho o ds of the input graph. W e then use the symmetry of its corresp onding co-bipartite incidence graph to argue that, besides a p olynomial num b er of solutions of b ounded size, the problem amoun ts to generating all maximal irredundan t sets of the input graph t wice, and thus obtain the aforemen tioned result. Finally , we obtain the following tractable result on strongly orderable graphs, which generalize c hordal bipartite graphs [Dra00]. Theorem 1.5. Ther e is a p olynomial-delay and p olynomial-sp ac e algorithm solving GraphMirr · Enum on str ongly or der able gr aphs. W e devise the algorithm of Theorem 1.5 b y using ordered generation in the trace of the closed-neighborho o d h yp ergraph. Our metho d is based on earlier work of [ CKMU19 ], where the algorithm aims to break any redundancies obtained from adding a new vertex in to the set by using structural prop erties of an elimination ordering. W e turn to minimal redundan t sets and obtain the following deﬁnite answ er to Question 1.2 for GraphMred · Enum . Theorem 1.6. Ther e is no output-p olynomial-time algorithm for GraphMred · Enum unless P  = NP , even when r estricte d to split and c o-bip artite gr aphs. Both results are obtained by reducing from HypMred · Enum and arguing that, except for a polynomial num b er of minimal redundant sets, the remaining solutions of the constructed instance of GraphMred · Enum are in an one-to-one corresp ondence with the solutions of the h yp ergraph. T o argue the existence of no more than a p olynomial n umber of solutions to discard, we exploit the structure of split graphs and, for the case of co-bipartite graphs, the fact that HypMred · Enum remains intractable when the h yp ergraph has dimension equals three. On the other hand, we obtain the following p ositive results. Let us note that ( C 3 , C 5 , C 6 , C 8 ) -free graphs con tain chordal bipartite graphs, which are ( C 3 , C ≥ 5 ) -free [GG78]. Theorem 1.7. Ther e is a p olynomial-delay algorithm that solves GraphMred · Enum on ( C 3 , C 5 , C 6 , C 8 ) -fr e e gr aphs, and an incr emental quasi-p olynomial-time algorithm that solves GraphMred · Enum on ( C 3 , C 5 , C 6 ) -fr e e gr aphs. These algorithms rely on a characterization of the minimal redundant sets of ( C 3 , C 5 , C 6 ) -free graphs. In particular, we show that the num b er of redundant v ertices in a minimal redundant set is b ounded for these classes of graphs and that, except for the case of a unique redundant vertex, the set either has b ounded size or corresponds to the closed neigh b orho o d of some vertex in the set. The remaining case where the set has a single redundant vertex is then reduced to instances of red-blue domination for which algorithms running in the sp eciﬁed times are known on these classes. 4 v ≤ ≤ ≤ u 1 u 2 u 3 u 4 Figure 1: A graph G in which the vertex v is quasi-simple. Within its neigh b orho o d, we ha ve u 1 ≤ u 2 ≤ u 3 ≤ u 4 where ≤ indicates comparabilit y . Organization. The rest of the pap er is organized as follows. W e introduce concepts in Section 2. W e fo cus on maximal irredundant sets in Section 3 that we break in to subsections according to the graph classes w e consider. W e then turn on minimal redundan t sets in Section 4 that we organize analogously . Lastly , future directions are discussed in Section 5. 2 Preliminaries In this pap er, w e consider undirected ﬁnite simple graphs. W e assume the reader is familiar with standard terminology from graph and hypergraph theories, and refer to [ Ber84 , Die05 ] for the notations not deﬁned b elo w. Given a vertex v and an integer d , b y N d ( v ) and N d [ v ] we resp ectively mean the set of vertices lying at distance exactly d and at most d from v . Strongly orderable graphs. Giv en a graph G , a pair of vertices x and y of G are c omp ar able if either N ( x ) \ { y } ⊆ N ( y ) \ { x } or N ( y ) \ { x } ⊆ N ( x ) \ { y } . A v ertex x is quasi-simple if the vertices in N ( x ) are pairwise comparable. In Figure 1, we give an example of a graph admitting a quasi-simple vertex. Lastly , a quasi-simple elimination or dering is an ordering v 1 , . . . , v n of V ( G ) such that for ev ery i ∈ [ n ] it follows that v i is quasi-simple in the graph G [ { v 1 , . . . , v i } ] induced by the i ﬁrst vertices. Graphs that admit a quasi-simple elimination ordering are precisely the str ongly or der able gr aphs , whic h were introduced as a generalization of both chordal bipartite and strongly chordal graphs in [Dra00]. Hyp ergraphs. Let H b e a hypergraph. Then, the v ertices and edges of H are denoted b y V ( H ) and E ( H ) , respectively . W e call inc H ( x ) : = { E ∈ E ( H ) | x ∈ E } the set of h yp eredges con taining x . The subscript is omitted when the hypergraph is clear from con text. If G is a graph, then its close d-neighb orho o d hyp er gr aph , denoted N ( G ) , is deﬁned on the same v ertex set and hyperedge set { N [ x ] | x ∈ V ( G ) } . Let E 1 , . . . , E m b e an ordering of E ( H ) . The tr ansp ose d hyp er gr aph of H is the h yp ergraph H t with V ( H t ) : = { e 1 , . . . , e m } and E ( H t ) : = { F 1 , . . . , F n } where F i = { e j ∈ 5 V ( H t ) | v i ∈ E j } . Moreov er, given a set S ⊆ V , the tr ac e of H with resp ect to S is the h yp ergraph H S where V ( H S ) = S and E ( H S ) : = { E ∩ S | E ∈ E ( H ) , E ∩ S  = ∅} . Incidence graphs. Let H b e a hypergraph with V ( H ) = { v 1 , . . . , v n } and E ( H ) = { E 1 , . . . , E m } . The incidenc e bip artite gr aph of H is the bipartite graph B ( H ) with parts V = V ( H ) and U : = { e 1 , . . . , e m } , and edge set { v i e j | v i ∈ E j } . Similarly , the incidenc e c o-bip artite gr aph of H is the co-bipartite graph C ( H ) obtained from B ( H ) by turning V and U in to cliques. Finally , w e can deﬁne tw o incidenc e split gr aphs for H dep ending on whic h of V and U is the indep endent set. W e deﬁne S 1 ( H ) to b e the split graph obtained from B ( H ) b y making U a clique, and S 2 ( H ) to b e the split graph where V is the clique. W e give in Figure 2 an example illustrating all four deﬁnitions. v 4 v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 H B ( H ) C ( H ) S 1 ( H ) S 2 ( H ) E 1 E 2 E 3 E 4 v 2 v 1 v 3 Figure 2: An hypergraph H and its incidence graphs B ( H ) , C ( H ) , S 1 ( H ) and S 2 ( H ) . In graphs, cliques are indicated b y dotted grey zones. Domination. Let G b e a graph. W e sa y that a subset of vertices D is a dominating set of G if V ( G ) = N [ D ] . Given a bipartition ( A, B ) of the vertices, we say that a subset D ⊆ A dominates B if B ⊆ N ( D ) . In this latter context, these sets are usually referred to as r e d-blue dominating sets , where elements of A are assumed to b e colored red, and those in B to b e blue. Generating every minimal red-blue dominating set is known to b e solv able with polynomial dela y and space when restricted to chordal bipartite graphs [ GHK + 16 , CDG25 ]. It can also b e solved with p olynomial delay in ( C 6 , C 8 ) -free bipartite graphs [ KKP18 ] as long as red and blue v ertices b elong to distinct parts of the graph. In general, the problem is equiv alen t to hypergraph dualization and can thus b e solv ed in incremental-quasi-polynomial time; see e.g. [EMG08, GHK + 16]. (Ir)redundancy . Let H b e a hypergraph, I ⊆ V ( H ) , and x ∈ I . W e call private e dge of x with r esp e ct to I the edges in the set p riv H ( x, I ) : = { E ∈ E ( H ) | E ∩ I = { x }} . If H is clear from the context, we drop it from this notation. W e say that I is an irr e dundant set of H if ev ery element x of I has a priv ate edge, and that it is a r e dundant set otherwise. 6 Observ e that, given H , one can chec k in p olynomial time whether a set I of vertices is redundan t or irredundan t b y computing p riv ( x, I ) for each x ∈ I . W e denote b y MaxIrr ( H ) the family of inclusion-wise maximal irredundant sets of H , and by MinRed ( H ) its family of inclusion-wise minimal redundan t sets. The problems of generating all maximal irredundant sets (resp. all minimal redundan t sets) th us read as follows: Maximal Irredundan t Sets Enumeration in Hyp ergraphs ( HypMirr · Enum ) Input: a h yp ergraph H . Output: the family MaxIrr ( H ) . Minimal Redundan t Sets Enumeration in Hyp ergraphs ( HypMred · Enum ) Input: a h yp ergraph H . Output: the family MinRed ( H ) . W e recall that these problems were sho wn intractable ev en on hypergraphs of dimension at most three [ BM16 , BM24 ] and refer to the in tro duction for the state of the art on these problems. In this pap er, we are interested in the analogues of redundancy and irredundancy in graphs, that w e recall no w. Let G b e a graph, I ⊆ V ( G ) , and x ∈ I . W e call private neighb or of x the elements of the set p riv G ( x, I ) : = { y ∈ N [ x ] | N [ y ] ∩ I = { x }} . Note that a vertex may b e self-priv ate. Analogously , as for hypergraphs, a subset I of vertices is called irr e dundant if p riv G ( x, I )  = ∅ for all x ∈ I , and r e dundant otherwise. W e denote b y MaxIrr ( G ) and MinRed ( G ) the set of maximal irredundan t and minimal redundant sets of G . T o av oid any confusion b etw een graphs and h yp ergraphs, we will only refer to h yp ergraphs using calligraphic letters. The en umeration problems we are interested in are deﬁned as follows. Maximal Irredundan t Sets Enumeration on Graphs ( GraphMirr · Enum ) Input: a graph G . Output: the family MaxIrr ( G ) . Minimal Redundan t Sets Enumeration on Graphs ( GraphMred · Enum ) Input: a graph G . Output: the family MinRed ( G ) . As stated in the introduction, it is easily seen that MaxIrr ( G ) and MinRed ( G ) coincide with the maximal irredundant and minimal redundant sets of N ( G ) . They should, ho wev er, not b e confused with the maximal irredun dan t and minimal redundant of G seen as a h yp ergraph of dimension 2. 3 Maximal irredundant sets In this section we c haracterize the complexit y of GraphMirr · Enum in split and co- bipartite graphs, as well as in strongly orderable graphs, whic h generalize chordal bipartite graphs. 7 3.1 Strongly orderable graphs W e give an algorithm that generates all maximal irredundant sets of strongly orderable graphs with p olynomial delay and space. Our algorithm is based on the or der e d gener ation framew ork (also known as the se quential metho d ) that has b een successfully applied to v arious instances admitting elimination orders [ EGM03 , BDMI24 , CDG25 ]. Our algorithm, how ever, diﬀers from the strategy used in the aforemen tioned w orks b y considering traces of the hypergraph (instead of its induced subhypergraphs) in the decomp osition, follo wing what has b een done in [CKMU19] for irredundant sets. Let G b e a graph. In the remaining, let us ﬁx an arbitrary ordering v 1 , . . . , v n of the v ertex set of G , and let V i : = { v 1 , . . . , v i } denote the set of its i ﬁrst v ertices. Let H : = N ( G ) be the closed-neighborho o d hypergraph of G , and H i : = H V i b e a shorthand for the trace of H on V i . F or each I ⊆ V i and x ∈ I , by p riv i ( x, I ) w e mean the set of priv ate edges of x with resp ect to I in H i , and by inc i ( x ) we mean the collection of h yp eredges of H i that con tain x . Note that MaxIrr ( H 1 ) = {{ v 1 }} . This is due to N [ v 1 ] b eing a h yp eredge in H , hence the singleton { v 1 } is the only h yp eredge of H 1 . The goal of the algorithm will b e to construct MaxIrr ( H i ) for i ranging from 1 to n , from which we only output MaxIrr ( H n ) = MaxIrr ( H ) . W e will call p artial solutions the maximal irredundant sets of H i for i < n , and solutions those of H n . Ho wev er, and to guaran tee p olynomial delay b etw een consecutive outputs, we will not construct the sets MaxIrr ( H i ) one after the other. Rather, we will deﬁne a tree ov er the set of solutions and partial solutions that will b e tra versed by our algorithm. T o deﬁne this tree, we start with an easy observ ation whose pro of is omitted; see also [ CKMU19 ]. The intuition is that whenever a v ertex v j has a priv ate edge in H i for some i > j , then it k eeps that priv ate edge in any subhypergraph H k , where j ≤ k ≤ i . Prop osition 3.1. L et i ∈ { 2 , . . . , n } and I ∈ MaxIrr ( H i ) . Then I \ { v i } is an irr e dundant set of H i − 1 . Let i ∈ { 2 , . . . , n } and I ∈ MaxIrr ( H i ) . The p ar ent of I with r esp e ct to i , denoted pa rent ( I , i ) , is deﬁned as the set obtained by the following pro cedure. First, if v i b elongs to I , we remov e it. Then, while there exists a vertex x ∈ V i − 1 \ I suc h that I ∪ { x } is irredundan t in H i − 1 , we add the smallest such vertex x in to I . Note that the obtained set is uniquely deﬁned. By Prop osition 3.1, it is an irredundan t set of H i − 1 after the ﬁrst step, and it is extended into a maximal irredundan t set of H i − 1 at the end of the pro cedure. By construction we thus get pa rent ( I , i ) ∈ MaxIrr ( H i − 1 ) and pa rent ( I , i ) = I whenever v i ∈ I . Con versely , giv en i ∈ [ n − 1] and I ⋆ ∈ MaxIrr ( H i ) , we deﬁne the childr en of I ⋆ with r esp e ct to i as children ( I ⋆ , i ) : = { I ∈ MaxIrr ( H i +1 ) | parent ( I , i + 1) = I ⋆ } . The follo wing shows that any partial solution has at least one child, and that it has precisely one c hild if it can b e extended in to an irredundant set of H i +1 b y adding v i +1 . Lemma 3.2. L et i ∈ [ n − 1] and I ⋆ ∈ MaxIrr ( H i ) . Then, either: • I ⋆ ∈ MaxIrr ( H i +1 ) , in which c ase I ⋆ ∈ children ( I ⋆ , i ) ; or • I ⋆ ∪ { v i +1 } ∈ MaxIrr ( H i +1 ) , in which c ase children ( I ⋆ , i ) = { I ⋆ ∪ { v i +1 }} . 8 Pr o of. Let I ⋆ ∈ MaxIrr ( H i ) . By deﬁnition, I ⋆ do es not contain v i +1 , and no vertex in V i can b e added to I ⋆ while maintaining the irredundancy of I ⋆ in H i . Hence the pa rent pro cedure, when p erformed on ( I ⋆ , i + 1) in case I ⋆ ∈ MaxIrr ( H i +1 ) , do es not mo dify the set and gives pa rent ( I ⋆ , i + 1) = I ⋆ , th us I ⋆ ∈ children ( I ⋆ , i ) . This prov es the ﬁrst assertion of the claim. Let us now assume that the ﬁrst case do es not arise, i.e., I ⋆ ∈ MaxIrr ( H i +1 ) . Supp ose, for the sake of con tradiction, that I ⋆ ∪ { v i +1 } is redundant. Since I ⋆ ∈ MaxIrr ( H i ) every set I ⋆ ∪ { x } with x ∈ V i \ I ⋆ is redundant in H i . The same is true in H i +1 since a priv ate edge of such a x in H i +1 w ould yield one (p ossibly reduced to its intersection with V i ) in H i . How ever, this implies that I ⋆ is inclusion-wise maximal with respect to b eing irredundan t in H i +1 , con tradicting our assumption. So I ⋆ ∪ { v i +1 } is irredundan t. As H i and H i +1 only diﬀer by the trace of the edges incident to v i +1 , the set I ⋆ ∪ { v i +1 } m ust b e maximal with that prop ert y . By deﬁnition of the paren t relation, we thus obtain { I ⋆ ∪ { v i +1 }} ⊆ children ( I ⋆ , i ) . The other inclusion follo ws from the fact that any I ′ ∈ children ( I ⋆ , i ) satisﬁes I ′ \ { v i +1 } ⊆ I ⋆ , hence I ′ ⊆ I ⋆ ∪ { v i +1 } . Note that the pa rent relation deﬁnes a tree T on the set of nodes { ( I , i ) | I ∈ MaxIrr ( H i ) , 1 ≤ i ≤ n } , whose ro ot is ( { v 1 } , 1) , and where there is an edge betw een tw o no des ( I ⋆ , i ) and ( I , i + 1) if I ∈ children ( I ⋆ , i ) . Moreo ver, b y Lemma 3.2, ev ery no de ( I , i ) corresp onding to a partial solution I has a child. Hence the set of leav es of this tree is precisely the family { ( I , n ) | I ∈ MaxIrr ( H ) } we wish to en umerate. In the following, we show that it is suﬃcient to b e able to list the c hildren with p olynomial dela y and space to derive a p olynomial delay and space algorithm for Graph Mirr · Enum . Prop osition 3.3. L et f , s : N → N b e two functions. Supp ose that ther e is an algorithm that, for any i ∈ [ n − 1] and I ⋆ ∈ MaxIrr ( H i ) , gener ates children ( I ⋆ , i ) with f ( n ) delay and s ( n ) sp ac e. Then ther e is an algorithm that gener ates MaxIrr ( H ) with O ( n · f ( n )) delay and O ( n · s ( n )) sp ac e. Pr o of. Let A b e the c hildren-generation algorithm as given by assumption. W e describ e an algorithm B whic h generates all maximal irredundant sets with the aforemen tioned w orst-case guarantees as follows. The algorithm p erforms a DFS on T starting from the ro ot ( { v 1 } , 1) and outputs each leaf as it is visited. Its correctness follo ws by the ab ov e discussion. Since ev ery leaf is at depth n , and for every i ∈ [ n − 1] and I ∈ MaxIrr ( H i ) the children of ( I , i ) can b e computed with f ( n ) delay , a ﬁrst maximal irredundant set of H is obtained in O ( n · f ( n )) time. As leav es are at a distance at most 2 n in T , the time sp ent by the DFS b etw een consecutiv e outputs is b ounded by O ( n · f ( n )) as well. Finally , after the last output, the algorithm concludes that no other solution exists within the same time after having bac ktrack ed to the ro ot and concluding that it has explored all its children. Note that the algorithm requires O ( n · s ( n )) space for maintaining the recursion stack for bac ktracking. This concludes the pro of. W e now fo cus on generating children ( I ⋆ , i ) for ﬁxed i ∈ [ n − 1] and I ⋆ ∈ MaxIrr ( H i ) suc h that I ⋆ ∪ { v i +1 } is redundant. Recall by Lemma 3.2 that I ⋆ is one of these children, and we may thus fo cus on those con taining v i +1 , which we shall call non-trivial childr en . 9 Our strategy is to ﬁnd all sets of the form Z : = ( I ⋆ \ X ) ∪ { v i +1 } , where X ⊆ I ⋆ , that are maximal irredundant sets of H i +1 . W e call each such set Z an extension of I ⋆ to i + 1 . Note that these extensions do not deﬁne sup ersets of I ⋆ , as they are obtained b y adding v i +1 and remo ving elements of I ⋆ . As another remark, note that p ossibly not all extensions of I ⋆ to i + 1 are children of I ⋆ with resp ect to i . How ev er, all non-trivial c hildren I of I ⋆ are extensions of I ⋆ to i + 1 : by deﬁnition, if I ⋆ = pa rent ( I ) then I ⋆ is of the form ( I \ { v i +1 } ) ∪ X for some X ⊆ V i , and so I = ( I ⋆ \ X ) ∪ { v i +1 } with X ⊆ I ⋆ . Let us now assume that G is strongly orderable and that v 1 , . . . , v n is a quasi-simple elimination ordering of its vertices. In the follo wing, by N i ( x ) we mean the neighborho o d of x in tersected with V i . T o compute all extensions eﬃciently , w e will rely on a characterization of their in tersection with the set of v ertices that b ecome redundan t when adding v i +1 to I ⋆ . More formally , let R ( I ⋆ , i ) : = { x ∈ I ⋆ | priv i +1 ( x, I ⋆ ) ⊆ inc i +1 ( v i +1 ) } denote suc h a set. Note that the vertices in R ( I ⋆ , i ) lie at a distance of at most 2 from v i +1 in G [ V i +1 ] . An upp er b ound on the n umber of elements in R ( I ⋆ , i ) found in an extension of I ⋆ is giv en in the follo wing lemma. Lemma 3.4. L et Z b e an extension of I ⋆ to i + 1 . Then: • | R ( I ⋆ , i ) ∩ N i +1 ( v i +1 ) ∩ Z | ≤ 1 ; and • | R ( I ⋆ , i ) ∩ N 2 i +1 ( v i +1 ) | ≤ 1 . Pr o of. By deﬁnition of R ( I ⋆ , i ) , the priv ate edges of an element x ∈ R ( I ⋆ , i ) with resp ect to I ⋆ in H i +1 all con tain v i +1 . By the deﬁnition of H , eac h such priv ate edge is thus of the form N i +1 [ u ] for some u ∈ N i +1 [ v i +1 ] , with p ossibly u = x , i.e., x is self-priv ate and lies in the neighborho o d of v i +1 . Recall that b y deﬁnition, Z is a maximal irredundant set of H i +1 , and so eac h of its elements has a priv ate edge in H i +1 . Assume, for the sake of contradiction, that there are at least tw o distinct vertices x 1 and x 2 in R ( I ⋆ , i ) ∩ N i +1 ( v i +1 ) ∩ Z . Since v i +1 is quasi-simple, x 1 and x 2 are pairwise comparable in G [ V i +1 ] . So the only wa y for x 1 and x 2 to b oth hav e a priv ate neighbor as describ ed ab ov e is for one to b e self-priv ate with resp ect to I ⋆ in H i +1 . How ev er, by deﬁnition, v i +1 b elongs to Z . So x 1 and x 2 cannot b e self-priv ate with resp ect to Z . W e conclude that one of the tw o is redundant in Z , which contradicts our assumption and pro ves the ﬁrst item. A dditionally , from the fact that v i +1 is quasi-simple, we also derive that for every pair of distinct vertices y 1 , y 2 ∈ N 2 i +1 ( v i +1 ) , either N i +1 ( y 1 ) ∩ N i +1 ( v i +1 ) ⊆ N i +1 ( y 2 ) ∩ N i +1 ( v i +1 ) or the opp osite holds. Consequen tly , a pair of v ertices in N 2 i +1 ( v i +1 ) can neither sim ultaneously hav e priv ate edges as describ ed ab o ve. Hence, they cannot b oth ha ve a priv ate edge in H i +1 , pro ving the second item of the claim. Let us p oint out that Z cannot b e remov ed from the ab ov e statement. In fact, the cardinalit y of R ( I ⋆ , i ) ∩ N i +1 ( v i +1 ) is un b ounded in general. This can b e seen b y considering the situation where N i +1 ( v i +1 ) is an indep endent set whose elements are self-priv ate with resp ect to I ⋆ . A dditionally , let us p oint that the set X ⊆ I ⋆ suc h that Z = ( I ⋆ \ X ) ∪ { v i +1 } is an extension may contain vertices that are not in R ( I ⋆ , i ) . Indeed, removing v ertices from I ⋆ \ R ( I ⋆ , i ) may provide priv ate edges to the remaining vertices in R ( I ⋆ , i ) . 10 Ho wev er, from Lemma 3.4 we know that at most tw o vertices from R ( I ⋆ , i ) may b e k ept in an extension, from which we derive the following. Lemma 3.5. The numb er of extensions of I ⋆ to i + 1 is b ounde d by O ( n 3 ) , and these extensions c an b e c ompute d in p olynomial time in n . Pr o of. By Lemma 3.4 each extension Z in tersects R ( I ⋆ , i ) on at most tw o vertices: one p ossibly b eing the unique vertex x in R ( I ⋆ , i ) ∩ N 2 i +1 ( v i +1 ) if it exists, and the other v ertex y b eing among those in R ( I ⋆ , i ) ∩ N i +1 ( v i +1 ) , if it exists. These vertices x and y ha ve at least one priv ate edge each. Note that all other vertices in Z are in I ⋆ \ R ( I ⋆ , i ) , hence ha ve priv ate edges with respect to I ⋆ . So the only reason for vertices in I ⋆ \ R ( I ⋆ , i ) not to b e included in Z is b ecause they participate to intersect all priv ate edges of x or y ; otherwise they could b e added to Z , contradicting its maximality . W e conclude that the set of all extensions Z can b e found by the follo wing pro cedure. First we decide whether to select the v ertex x ∈ R ( I ⋆ , i ) ∩ N 2 i +1 ( v i +1 ) , if it exists, and then select at most one, p ossibly none, vertex y ∈ R ( I ⋆ , i ) ∩ N i +1 ( v i +1 ) . F or each of the at most t wo selected v ertices x and y , w e choose one of their incident edges E x ∈ inc i +1 ( x ) and E y ∈ inc i +1 ( y ) to b ecome their resp ectiv e priv ate edges. The ﬁrst t wo steps amount to remo ving R ( I ⋆ , i ) from I ⋆ except for the p ossibly selected vertices x and y . The second step is to remov e ( E x \ { x } ) ∪ ( E y \ { y } ) from I ⋆ . F or each obtained set w e chec k whether it is a maximal irredundan t set and output it if this is the case. Note that the total num b er of diﬀerent selections is b ounded by O ( n 3 ) and that this pro cedures indeed runs in p olynomial time in the n umber of vertices. W e are ready to show that generating the c hildren of a partial solution can b e done with p olynomial dela y and space, by enumerating all extensions. Lemma 3.6. Ther e is an algorithm that gener ates children ( I ⋆ , i ) in p olynomial time. Pr o of. By Lemma 3.2, tw o cases arise. Either I ⋆ ∪ { v i +1 } ∈ MaxIrr ( H i +1 ) , in which case w e output it and are done with the en umeration, or I ⋆ ∈ MaxIrr ( H i +1 ) , in which case we output it and are left with the generation of the non-trivial children of I ⋆ . T o enumerate them, we start enumerating all extensions in p olynomial time using Lemma 3.5. F or ev ery extension, we chec k whether it is a c hild of I ⋆ b y running the pa rent pro cedure and output it if it is p ositiv e. As every step takes p olynomial time and since I ⋆ has a p olynomial n umber of children by Lemma 3.5, we obtain the desired result. W e conclude to Theorem 1.5 that w e restate here as a corollary of Prop osition 3.3 and Lemma 3.6. Theorem 1.5. Ther e is a p olynomial-delay and p olynomial-sp ac e algorithm solving GraphMirr · Enum on str ongly or der able gr aphs. 3.2 Co-bipartite graphs In this section, w e show that GraphMirr · Enum on co-bipartite graphs is as hard as GraphMirr · Enum on arbitrary graphs. Our reduction considers the co-bipartite incidence graph C ( N ( G )) of the closed-neighborho o d hypergraph N ( G ) of a given graph G ; see Figure 3 for an illustration. Using the fact that C ( N ( G )) is symmetric, we prov e 11 v 1 u 1 u 2 u 3 u 4 v 1 v 5 u 5 N [ v 5 ] G C ( N ( G )) v 5 v 2 v 2 v 3 v 4 v 3 v 4 Figure 3: The reduction of Theorem 1.4. On the left a graph G where shaded vertices indicate a maximal irredundan t set. On the right the incidence co-bipartite graph C ( N ( G )) of N ( G ) . Grey dotted zones indicate the clique bipartition and the shaded v ertices form the maximal irredundant set C ( N ( G )) induced by the maximal irredundant set in pictured in G . The vertices b oxed in blue illustrate how N [ v 5 ] (in G ) is enco ded in C ( N ( G )) . that listing the maximal irredundant sets of C ( N ( G )) amounts to list each maximal irredundan t set of G t wice, plus a p olynomial n umber of other solutions of size 2 spread across the parts of C ( N ( G )) . Theorem 1.4. Ther e is an output-p olynomial-time algorithm solving GraphMirr · Enum on gener al gr aphs if and only if ther e is one solving the same pr oblem on c o-bip artite gr aphs. Pr o of. The if part follo ws from the fact that an algorithm for GraphMirr · Enum for arbitrary graphs can b e used in particular on a co-bipartite graph. Let us thus show the only if part. Let G b e a non-empty graph with vertex set V = { v 1 , . . . , v n } . Let us consider the incidence co-bipartite graph C : = C ( N ( G )) of the closed-neighborho o d hypergraph N ( G ) of G . The t wo parts of C are V and U = { u 1 , . . . , u n } where u i represen ts N [ v i ] . Note that C is symmetric: w e hav e u i v j ∈ E ( C ) if and only if u j v i ∈ E ( C ) , for all i, j ∈ { 1 , . . . , n } . Moreov er, the incidences betw een U and V coincide with those in G : w e ha ve u i v j ∈ E ( C ) if and only if v i v j ∈ E ( G ) , for all i  = j ∈ { 1 , . . . , n } . F or any I ⊆ V , let us denote b y U ( I ) : = { u i | v i ∈ I } the “copy” of I in U . W e shall ﬁrst prov e that MaxIrr ( C ) = X ∪ I 1 ∪ I 2 with X : = { X | X ∈ MaxIrr ( C ) , | X | ≤ 2 } I 1 : = { I | I ∈ MaxIrr ( G ) , | I | ≥ 3 } I 2 : = { U ( I ) | I ∈ MaxIrr ( G ) , | I | ≥ 3 } where the sets X , I 1 , and I 2 are purp osely reﬁned b y cardinality in order to av oid unnecessary tec hnicalities in the pro of. Let us ﬁrst prov e the inclusion MaxIrr ( C ) ⊆ X ∪ I 1 ∪ I 2 . Let I ∈ MaxIrr ( C ) . Let us further assume that | I | ≥ 3 as the inclusion trivially holds for smaller I . Note that I is either a subset of V , or a subset of U , as otherwise it contains a prop er dominating subset, and th us would b e redundant. 12 If I ⊆ V , since | I | ≥ 3 , no elemen t of I has a priv ate neighbor in V . Hence I is maximal with the prop erty that its elements hav e priv ate neigh b ors in U . No w, since v i u j ∈ E ( C ) if and only if v i v j ∈ E ( G ) , I is actually maximal with the p rop ert y that its elemen ts hav e priv ate neighbors in G . Th us I ∈ I 1 . The case I ⊆ U leads to the conclusion that I ∈ I 2 b y symmetric arguments. Let us now prov e the other inclusion. Let I ∈ I 1 . As v i u j ∈ E ( C ) if and only if i = j or v i v j ∈ E ( G ) , every element of I has a priv ate neighbor in U . So I is irredundant in C , and by the maximalit y of I in G , adding any vertex v ∈ V \ I to I mak es an elemen t of I ∪ { v } redundan t with resp ect to U . A dding a vertex u of U to I w ould make all v ertices of I redundan t with resp ect to U , since U is a clique. No w, since | I | ≥ 3 , no elemen t of I has a priv ate in V . W e conclude that I is a maximal irredundan t set of C . The case of I ∈ I 2 leads to I ∈ MaxIrr ( C ) b y symmetric arguments. W e are now ready to conclude the pro of. Assume there is an algorithm A solving GraphMirr · Enum on any co-bipartite graph C in output-p olynomial time. W e give an output-p olynomial time algorithm solving GraphMirr · Enum for an y graph G . First the algorithm outputs all members of MaxIrr ( G ) of size at most 2 by ranging ov er subsets of size 2 and testing if they deﬁne a maximal irredundant set of G . Then, the algorithm builds C , runs A on C , and outputs only those mem b ers of MaxIrr ( G ) of size at least 3. These steps are done in a time which is p olynomial in the sizes of C and MaxIrr ( C ) . Due to the ab ov e equality , each solution will b e correctly output. Again using this equality , w e hav e | MaxIrr ( C ) | ≤ 2 × | MaxIrr ( G ) | + | V ( G ) | 2 . Therefore, the whole algorithm runs in polynomial time in the sizes of G and MaxIrr ( G ) as exp ected. This concludes the pro of. W e note that the reduction in the pro of of Theorem 1.4 actually preserves incremental p olynomial time, since the procedure ma y list linearly man y solutions that corresp ond to already obtained solutions b efore outputting a new solution. 3.3 Split graphs Finally , for completeness, we include a pro of of Theorem 1.3 which was initially claimed b y Uno in [BBHK15]. Lemma 3.7. The maximal irr e dundant sets of a split gr aph G ar e pr e cisely its minimal dominating sets. Pr o of. Recall that in any graph, ev ery minimal dominating set is a maximal irredundant set; see Section 2. W e show that the con verse is true in split graphs. Let G b e a split graph of clique-indep endent set partition ( C, S ) , and I ∈ MaxIrr ( G ) . Let us assume without loss of generality that S is maximized in this partition, i.e., every element of C has a neighbor in S . Note that for any y ∈ I ∩ S , we hav e that I ∩ N ( y ) = ∅ , as otherwise y is redundan t. Consequently , every element of I ∩ S is self-priv ate, and every element of I ∩ C has a priv ate neigh b or in S . Supp ose tow ard a contradiction that I is not a dominating set. If a vertex y of the indep enden t set is not dominated, then y is self-priv ate with resp ect to I ∪ { y } . Moreov er, b y the ab ov e discussion, every other vertex in I either is self-priv ate if it b elongs to S , or has a priv ate neighbor in S . So I ∪ { y } is irredundant, contradicting the maximality of I . On the other hand, if a vertex x of the clique is not dominated, then it has a neighbor y 13 in the indep endent set that is not dominated. Conducting the same argument as b efore, I ∪ { y } is irredundant, a contradiction. W e conclude to Theorem 1.3 using a linear delay and space algorithm for minimal dominating sets en umeration in split graphs, as given in [KLMN14]. 4 Minimal redundant sets In this section, we in vestigate the complexit y of GraphMred · Enum . W e ﬁrst prop ose a p olynomial-dela y algorithm for the class of ( C 3 , C 5 , C 6 , C 8 ) -free graphs. Recall that this class contains chordal bipartite graphs, whic h are precisely ( C 3 , C ≥ 5 ) -free graphs [ GG78 ]. Then, w e prov e that the problem is intractable for co-bipartite and split graphs. 4.1 Graphs excluding small cycles In this section, w e c haracterize the minimal redundant sets of graphs excluding small cycles. Our characterization relies on the n umber of redundant vertices in a minimal redundan t set and in volv es minimal red-blue dominating sets within a ball of radius 2. This yields a p olynomial-delay algorithm for GraphMred · Enum on ( C 3 , C 5 , C 6 , C 8 ) -free graphs—whic h generalize chordal bipartite graphs—relying on existing results from the literature on red-blue domination, and an output quasi-p olynomial-time algorithm on ( C 3 , C 5 , C 6 ) -free graphs. The remainder of the section is organized as follows. W e start by stating preliminary prop erties in general graphs in Section 4.1.1, and then turn to graphs with no sm all cycles. W e prov e prop erties for these graphs in Section 4.1.2, provide the aforemen tioned c haracterization of minimal redundant sets in Section 4.1.3, and describ e the algorithm in Section 4.1.4. 4.1.1 Preliminary prop erties in graphs If R is a redundant set, we denote b y red ( R ) : = { x ∈ R | priv ( x, R ) = ∅} the set of its redundan t vertices. The next lemma states that in a minimal redundan t set R with redundan t vertex x , any vertex other than x prev ents x from having a dedicated priv ate neigh b or. Lemma 4.1. L et G b e a gr aph and R ∈ MinRed ( G ) . Then for any two distinct vertic es x, y ∈ R such that x ∈ red ( R ) ther e exists a vertex z ∈ p riv ( x, R \ { y } ) ∩ priv ( y , R \ { x } ) . Pr o of. By the minimality of R , the vertex y satisﬁes p riv ( x, R \ { y } )  = ∅ . Let z ∈ p riv ( x, R \ { y } ) . Because z is a priv ate neighbor of x in R \ { y } and x is redundant in R , it follo ws that z is adjacen t to y , nonadjacen t to R \ { x, y } , and th us z ∈ p riv ( y , R \ { x } ) . This concludes the pro of. The minimalit y of the redundant sets also implies the following. Prop osition 4.2. L et G b e a gr aph and R ∈ MinRed ( G ) b e a minimal r e dundant set. Then R ⊆ N 2 [ x ] for every x ∈ red ( R ) . 14 4.1.2 Prop erties in graphs excluding small cycles Lemma 4.3. L et G b e a ( C 3 , C 5 ) -fr e e gr aph. Then for every vertex x ∈ V ( G ) , the gr aph induc e d by N 2 [ x ] is bip artite. Pr o of. Let x ∈ V ( G ) . As G is C 3 -free, N ( x ) is an indep endent set. Let us supp ose to ward a contradiction that there are tw o adjacent vertices z 1 , z 2 in N 2 ( x ) . Let y 1 and y 2 b e neigh b ors of z 1 and z 2 in N ( x ) , respectively . As G is C 3 -free, y 1  = y 2 . But then { z 1 , y 1 , x, y 2 , z 2 } induces a cycle of length 5 , a contradiction. So N 2 ( x ) induces an indep enden t set and the statement follows. In the following, w e show that a minimal redundant set con taining a redundant v ertex x and one of its neighbor y suc h that N ( y )  = { x } ma y contain other elements in N ( y ) , or in N ( x ) , but not in b oth sets at the same time, while it should intersect exactly one if y is redundan t. This is illustrated in Figure 4. Lemma 4.4. L et G b e a ( C 3 , C 5 ) -fr e e gr aph, R ∈ MinRed ( G ) b e a minimal r e dundant set, and x ∈ red ( R ) b e a r e dundant vertex. If y ∈ R is adjac ent to x and N ( y )  = { x } , then at most one of the fol lowing holds: • R ∩ ( N ( y ) \ { x } )  = ∅ ; or • R ∩ ( N ( x ) \ { y } )  = ∅ . Mor e over, if y ∈ red ( R ) then exactly one of the items is satisﬁe d. Pr o of. By con tradiction. Assume b oth items hold. The second item implies that x do es not b ecome self-priv ate after remo ving a vertex from R ∩ N ( x ) . F rom the ﬁrst item, the vertex y do es not b ecome a priv ate neigh b or of x in R \ { y } . Since G is C 3 -free, the v ertices x and y do not share a common neighbor, and thus x ∈ red ( R \ { y } ) , a con tradiction to the minimality of R . It remains to show that at least one of the items holds when y ∈ red ( R ) . If y ∈ red ( R ) the set R \ { y } minimally dominates N [ y ] . Let z ∈ N ( y ) \ { x } and assume that z / ∈ R , as otherwise the ﬁrst item holds. Then there exists w ∈ R \ { y } ∩ N ( z ) . By Prop osition 4.2, R ⊆ N 2 [ x ] . By Lemm a 4.3, w ∈ N ( x ) and since w  = y , the second item holds. This concludes the pro of. Lemma 4.5. L et G b e a ( C 3 , C 5 ) -fr e e gr aph, R ∈ MinRed ( G ) b e a minimal r e dundant set, and supp ose ther e exists an e dge xy such that x, y ∈ red ( R ) . Then either R = N [ y ] or R = N [ x ] . Pr o of. By Prop osition 4.2, R ⊆ N 2 [ x ] ∩ N 2 [ y ] . By Lemma 4.3, N 2 [ x ] ∩ N 2 [ y ] = N [ x ] ∪ N [ y ] , and thus N 2 ( x ) ∩ R = R ∩ ( N ( y ) \ { x } ) . By Lemma 4.4, either R ∩ ( N ( x ) \ { y } )  = ∅ or R ∩ ( N ( y ) \ { x } )  = ∅ . Without loss of generalit y , assume w e are in the ﬁrst case. By Lemma 4.4 we derive N 2 ( x ) ∩ R = ∅ . Since x is redundant, we get that N [ x ] ⊆ R , hence that N [ x ] = R by minimality of R . W e contin ue by proving prop erties on ( C 3 , C 5 , C 6 ) -free graphs. 15 N ( y ) \ { x } N ( x ) \ { y } y y N ( x ) \ { y } N ( y ) \ { x } N ( x ) \ { y } N ( y ) \ { x } N ( x ) \ { y } N ( y ) \ { x } y x x x x y Figure 4: The situations of Lemma 4.4. In each case, a minimal redundan t set is indicated b y shaded vertices with (yello w) b old vertices b eing redundant. The top-left, top-right, and b ottom-left cases illustrate that R in tersects at most one of N ( x ) \ { y } or N ( y ) \ { x } . The b ottom-right graph illustrates one of the tw o cases of Lemma 4.5, where y is also redundan t: R = N [ y ] . The other case, not pictured here, is R = N [ x ] . Lemma 4.6. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph, R ∈ MinRed ( G ) a minimal r e dundant set, x ∈ red ( R ) a r e dundant vertex, and z 1 , z 2 b e two distinct vertic es in R ∩ N 2 ( x ) . Then ( N ( z 1 ) ∩ N ( z 2 )) \ N ( x ) = ∅ . Pr o of. Let z 1 , z 2 b e t wo distinct vertices in R ∩ N 2 ( x ) . By Lemma 4.1, we derive that for each i ∈ [2] we hav e a vertex y i ∈ R suc h that y i ∈ priv ( x, R \ { z i } ) . By Lemma 4.3, the set N 2 [ x ] induces a bipartite graph. Now assuming there exists a v ertex w ∈ ( N ( z 1 ) ∩ N ( z 2 )) \ N ( x ) w e obtain an induced cycle of length 6 , a contradiction. This concludes our pro of. Lemma 4.7. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph. If R ∈ MinRed ( G ) and x ∈ red ( R ) , then: • | red ( R ) ∩ N ( x ) | ≤ 2 ; and • | red ( R ) ∩ N 2 ( x ) | ≤ 1 . Pr o of. Supp ose, for the sak e of contradiction, that S : = { y 1 , y 2 , y 3 } ⊆ red ( R ) ∩ N ( x ) . By Lemma 4.1, for each pair of indices i, j ∈ [3] , where i  = j , there exists a vertex z ij suc h that z ij ∈ p riv ( y i , R \ { y j } ) ∩ priv ( y j , R \ { y i } ) . In particular, z ij is nonadjacent to the remaining vertex in S \ { y i , y j } . By Lemma 4.3, the set S and the set of all such z ij are indep enden t. W e conclude that G con tains C 6 as an induced subgraph, a contradiction. This concludes the pro of for the ﬁrst item. 16 No w assume | red ( R ) ∩ N 2 ( x ) | ≥ 2 and let z 1 and z 2 b e t wo distinct vertices in this set. Recall that by Lemma 4.1 there exists a vertex w ∈ priv ( z 1 , R \ { z 2 } ) ∩ priv ( z 2 , R \ { z 1 } ) . This, together with the fact that N 2 ( x ) induces an indep enden t set by Lemma 4.3, implies that w / ∈ N 2 [ x ] . How ev er, this is not p ossible b ecause by Lemma 4.6 the vertices z 1 and z 2 do not share a common neighbor outside the neighborho o d of x . This concludes our pro of. F rom Lemma 4.7 w e conclude that if G is a ( C 3 , C 5 , C 6 ) -free graph, then the maxim um n umber of redundan t vertices a minimal redundan t set contains is 4. This b ound can b e further improv ed b y analyzing what happ ens when there exists a redundant vertex at distance t wo from another redundant vertex. Lemma 4.8. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph. If R ∈ MinRed ( G ) such that x ∈ red ( R ) and | red ( R ) ∩ N 2 ( x ) | = 1 , then | R ∩ N ( x ) | = 1 , and so | R | = 3 . Pr o of. Let red ( R ) ∩ N 2 ( x ) = { z } . If N ( z ) ⊆ N ( x ) , then b ecause z is redundant, there exists a v ertex y ∈ R ∩ N ( z ) . How ever, since x ∈ R the v ertex z is redundan t in { x, y , z } and, by minimality , we derive that R = { x, y , z } as desired. On the other hand, if N ( z ) ⊆ N ( x ) , then there exists a vertex w ∈ N ( z ) \ N ( x ) . Due to Lemma 4.3 the v ertex w is at a distance 3 of x and th us it do es not b elong to the redundant set R as a consequence of Prop osition 4.2. In addition, Lemma 4.6 establishes that no other v ertex in R ∩ N 2 ( x ) can b e adjacent to w , whic h mak es w a priv ate neigh b or of z and contradicts our assumption. This concludes our pro of. W e now conclude the following. Corollary 4.9. If G is ( C 3 , C 5 , C 6 ) -fr e e and R ∈ MinRed ( G ) , then R c ontains at most 3 r e dundant vertic es. The bound in Corollary 4.9 is tigh t, as witnessed b y 3 consecutiv e v ertices in a cycle of length 4 . W e end this section with a last prop erty that will b e used in the c haracterization of minimal redundan t sets. Lemma 4.10. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph. If R ∈ MinRed ( G ) such that x ∈ red ( R ) and | red ( R ) ∩ N ( x ) | = 2 , then | R | = 3 . Pr o of. Let red ( R ) ∩ N ( x ) = { y 1 , y 2 } b e the pair of redundan t vertices adjacent to x . W e pro ceed by contradiction. Supp ose that y 3 ∈ R ∩ N ( x ) \ { y 1 , y 2 } . It follo ws from Lemma 4.1 that for every pair of indices i, j ∈ [3] , where i  = j , there exists a vertex z ij suc h that z ij ∈ priv ( y i , R \ { y j } ) . By Lemma 4.3, the set { y 1 , z 12 , y 2 , z 23 , y 3 , z 13 } induces a cycle of length 6 , and w e arrive at a contradiction. It remains to show that R ∩ N 2 ( x ) is empty , and thus supp ose that z ∈ R ∩ N 2 ( x ) . By Lemma 4.3 the set N ( z ) ∩ N ( y i ) is empt y for each i ∈ [2] , and therefore by Lemma 4.1 w e must ha ve z ∈ p riv ( y i , R \ { z } ) . This leads to a contradiction, as z cannot satisfy this prop ert y for b oth vertices, and we conclude our pro of. 17 4.1.3 Characterization of redundan t sets in graphs with no small cycles In the remainder of this section, w e assume G to b e a ( C 3 , C 5 , C 6 ) -free graph. By Corollary 4.9 w e know that the num b er of redundan t vertices in a minimal redundant set of G is at most 3. In the follo wing, we c haracterize the minimal redundant sets dep ending on whether their n umber of redundant vertices is 3, 2, or 1. Let us recall that any minimal redundant set R con tains a redundant vertex x , and that by Lemma 4.7, it contains at most tw o additional redundan t vertices at distance 1 from x , and at most one at distance 2. W e deriv e the follo wing as a corollary of Lemmas 4.8 and 4.10. Corollary 4.11. If R ∈ MinRed ( G ) c ontains 3 r e dundant vertic es, then | R | = 3 . F or minimal redundan t sets containing tw o redundan t vertices, we derive the following as a corollary of Lemmas 4.5 and 4.8. Corollary 4.12. If R ∈ MinRed ( G ) c ontains 2 r e dundant vertic es, then either: • R = N [ x ] for some x ∈ red ( R ) ; or • | R | = 3 . W e are no w left with the characterization of minimal redundan t sets con taining exactly one redundant vertex which we call x . W e distinguish tw o cases dep ending on whether they intersect the neighborho o d of x on one or more vertices. These lemmas are b etter understo o d with accompanied Figure 5. Lemma 4.13. L et x ∈ V ( G ) and y ∈ N ( x ) . The fol lowing ar e e quivalent: (a) R ∈ MinRed ( G ) wher e red ( R ) = { x } and N ( x ) ∩ R = { y } . (b) R = S ∪ { x, y } wher e: i) S ⊆ N 2 ( x ) ; ii) The vertex y has a private neighb or with r esp e ct to S ∪ { x } ; iii) F or every z ∈ S satisfying N ( z ) ⊆ N ( x ) it fol lows that z / ∈ N ( y ) ; and iv) S minimal ly dominates N ( x ) \ { y } ; Pr o of. Supp ose that Item (a) holds. By Prop osition 4.2 the set R is con tained in N 2 [ x ] , pro ving Item (bi) . Because x is the only redundant vertex in R it follo ws that p riv ( y , R ) is non empty , pro ving Item (bii) . Moreo ver, w e deriv e that eac h v ertex z ∈ R \ { x, y } suc h that N ( z ) ⊆ N ( x ) must b e self-priv ate, and so in particular z / ∈ N ( y ) . This prov es Item (biii) . Lastly , S = R \ { x, y } minimally dominates N ( x ) \ { y } , as if it was not dominating, x w ould hav e a priv ate neighbor, and if it w as not minimal with that prop erty , there w ould exist a set S ′ ⊊ S suc h that S ′ dominates N ( x ) \ { y } , a contradiction to the minimalit y of R . This prov es Item (biv), hence the ﬁrst direction. Let us now assume that Item (b) holds. By Item (biv) and the fact that y ∈ R , the v ertex x is redundant. Moreov er, by the minimality of S , the vertex x gains a priv ate neigh b or in an y prop er subset of R con taining { x, y } . The same is true if we remo ve y from R , as x b ecomes self-priv ate in that case. Hence R is minimal with respect to 18 ha ving x redundan t, and it remains to argue that red ( R ) = { x } . W e do this by showing that ev ery element in R \ { x } has a priv ate neighbor. First, y has a priv ate neighbor by Item (bii) . Now, recall that by Lemma 4.3 N 2 [ x ] induces a bipartite graph, hence that the set S is an indep endent set. So the elements of S \ N ( y ) are self priv ate. It remains to consider the vertices z ∈ S ∩ N ( y ) . By Item (biii) the set N ( z ) \ N ( x ) is non-empty . Moreov er, by Lemma 4.6 no pair of vertices in S share a common neighbor outside N ( x ) , and so every element in the set N ( z ) \ N ( x ) is a priv ate neighbor of z . W e get that x is the only redundan t vertex of R as desired. This concludes the second direction. Lemma 4.14. L et x ∈ V ( G ) and Y ⊆ N ( x ) such that | Y | ≥ 2 . The fol lowing ar e e quivalent: (a) R ∈ MinRed ( G ) wher e red ( R ) = { x } and N ( x ) ∩ R = Y . (b) R = S ∪ Y ∪ { x } such that: i) S ⊆ N 2 ( x ) ; ii) S ∩ N ( Y ) = ∅ ; iii) Each vertex y ∈ Y has a private neighb or with r esp e ct to Y ∪ { x } ; iv) S minimal ly dominates N ( x ) \ Y . Pr o of. Supp ose that Item (a) holds and let S = R \ ( Y ∪ { x } ) . By Prop osition 4.2 R ⊆ N 2 [ x ] and w e obtain Item (bi) . Because x is the only redundant vertex in R , we ha ve N ( y )  = { x } for every y ∈ Y . Thus, b y Lemma 4.4, since | Y | ≥ 2 , for every y ∈ Y w e ha ve that R ∩ ( N ( y ) \ { x } ) = ∅ . This establishes Item (bii) . A dditionally , because eac h y is irredundant and adjacent to x , it follows that N ( y ) \ N ( Y \ { y } ) is non-empty and w e conclude Item (biii) . W e argue that S minimally dominates N ( x ) \ Y . If S w as not dominating, then x w ould hav e a priv ate neigh b or, con tradicting the redundancy of x ; and if S w as not minimal, then there w ould exist a set S ′ ⊊ S that dominates N ( x ) \ Y , and thus S ′ ∪ Y ∪ { x } w ould yield a smaller included redundant set, contradicting the minimalit y of R . This prov es Item (biv) and concludes the ﬁrst direction. Assume that Item (b) holds. By Item (biv) , the v ertex x is redundan t in the set R = S ∪ Y ∪ { x } . This set is minimal with that property as S minimally dominates N ( x ) \ Y b y Item (biv) , and removing an y element in Y w ould provide a priv ate neighbor to x , since Y is b oth an indep enden t set by Lemma 4.3, and nonadjacent to S b y (bii) . It remains to prov e that R do es not contain other redundant v ertices, i.e., that ev ery vertex in R \ { x } has a priv ate-neigh b or. Item (bii) and Lemma 4.3 show that the v ertices in S are self-priv ate. By Item (biii) eac h vertex y ∈ Y has a neighbor nonadjacent to vertices in Y \ { y } , and thus Lemma 4.3 establishes that these neighbors must b e priv ate since they cannot b e adjacen t neither to x nor to an element in S . 4.1.4 Algorithm for graphs with no small cycles W e now propose an algorithm for listing the minimal redundan t sets of a graph with no small cycles. It runs with polynomial-delay for ( C 3 , C 5 , C 6 , C 8 ) -free graphs and incremen tal quasi-p olynomial time for ( C 3 , C 5 , C 6 ) -graphs. 19 Y z ⋆ S x x y N ( x ) \ { y } N ( x ) \ Y S Figure 5: Illustration of Lemma 4.13 on the left, and Lemma 4.14 on the righ t. Shaded v ertices indicate minimal redundan t sets, and x (in b old yello w) is the redundan t vertex in each case. On the left, y (b o xed in green) has a priv ate vertex as well as each vertex of S adjacen t to y (the vertices of S are b oxed in red) and S minimally dominates N ( x ) \ { y } (b o xed in blue). On the right, Y has at least tw o elements, S con tains no vertex adjacent to Y and minimally dominates N ( x ) \ Y . The algorithm follo ws the breakdown of solutions based on redundant vertices we dev elop ed so far. First, following Corollaries 4.11 and 4.12, it enumerates all minimal redundan t s ets with 2 or 3 redundan t vertices by running through all p ossible closed neigh b orho o ds and triplets of v ertices, and chec king for each of these sets if it is a minimal redundan t, all of which in p olynomial time. Then it lists, for each vertex x , the minimal redundan t sets whose unique redundant vertex is x . According to Lemma 4.13 and Lemma 4.14, these can be further partitioned with resp ect to the num b er of neighbors of x in the redundan t sets. W e detail our approaches for b oth cases b elow. Let us ﬁrst consider the computation of all minimal redundant sets containing exactly one neighbor of x . The algorithm ﬁrst ranges o ver N ( x ) and for each c hoice y ∈ N ( x ) , selects a neigh b or z ⋆ of y that will be kept as a priv ate neigh b or of y . F or simplicity , let us put Z y : = { z | z ∈ N ( y ) , N ( z ) ⊆ N ( x ) } . Then, if an y , we list the minimal dominating sets S ⊆ N 2 ( x ) \ ( { z ⋆ } ∪ Z y ) of N ( x ) \ { y } . According to Lemma 4.13, the sets R = S ∪ { x, y } obtained that w ay are precisely the minimal redundant sets such that red ( R ) = { x } , N ( x ) ∩ R = { y } . Also, note that the sets S are the solutions to an instance of red-blue domination where we intend to minimally dominate N ( x ) \ { y } (blue) with v ertices from N 2 ( x ) \ ( { z ⋆ } ∪ Z y ) (red). As for the complexit y , the pairs { y , z ⋆ } can b e listed in p olynomial time, muc h as the set N 2 ( x ) \ ( { z ⋆ } ∪ Z y ) for each pair. Ho wev er, given some y , diﬀerent z ⋆ ’s can pro duce rep etitions, as these are chosen to b e excluded from the resulting solutions and hence can lead to same sets S . A given solution may thus b e rep eated O ( n ) times. In general graphs red-blue domination can b e solved in incremen tal quasi-p olynomial time [ FK96 , GHK + 16 ]. When applying the algorithm for red-blue domination on eac h z ⋆ successiv ely , the delay b efore the next solution th us remains b ounded by the num b er of solutions already obtained times a p olynomial factor (the num b er of rep etitions). W e obtain: 20 Prop osition 4.15. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph and let x ∈ V ( G ) . Ther e is an incr emental-quasi-p olynomial time algorithm that lists al l minimal r e dundant sets R wher e red ( R ) = { x } and R ∩ N ( x ) = { y } for some y ∈ N ( x ) . In ( C 6 , C 8 ) -free bipartite graphs, red-blue domination can b e solv ed with p olynomial dela y [ KKP18 ] when red and blue vertices belong to distinct parts. How ev er, our approach do es not directly preserve delay due to rep etitions. Y et, we can still preserve p olynomial dela y ov erall. T o do so, instead of outputting a previously unseen solution, we store it in to a queue, a member of which is popp ed and printed every O ( n ) · f ( n ) time, where f ( n ) is the delay for red-blue domination. Storing solutions into the queue and k eeping a structure to trac k rep etitions requires how ever exp onential space. W e get: Prop osition 4.16. L et G b e a ( C 3 , C 5 , C 6 , C 8 ) -fr e e gr aph and let x ∈ V ( G ) . Ther e is a p olynomial delay algorithm that lists al l minimal r e dundant sets R wher e red ( R ) = { x } and R ∩ N ( x ) = { y } for some y ∈ N ( x ) . W e now turn our attention to minimal redundan t sets containing at least tw o neighbors of x , corresp onding to Lemma 4.14. T wo steps are neede d. The ﬁrst step consists in iden tifying subsets of N ( x ) that can b e extended in to a minimal redundant sets. Giv en Y ⊆ N ( x ) , | Y | ≥ 2 , we sa y that Y is extendable with resp ect to x if there exists a minimal redundant set R with red ( R ) = { x } and R ∩ N ( x ) = Y . Below, we characterize extendable sets and show that they constitute an indep endence set system (i.e., a family of sets containing the empty set and b eing closed by subset), disregarding singletons and the empt y set. Lemma 4.17. L et x ∈ V ( G ) b e a vertex and Y ⊆ N ( x ) b e a set such that | Y | ≥ 2 . Then, the set Y is extendable with r esp e ct to x if and only if the two c onditions hold: (a) Each y ∈ Y has a private neighb or with r esp e ct to Y ∪ { x } ; and (b) The set N 2 ( x ) \ N ( Y ) dominates N ( x ) \ Y . Pr o of. Supp ose Y is extendable. Then, there is a set R ∈ MinRed ( G ) such that R ∩ N ( x ) = Y and x ∈ red ( R ) is the unique redundant vertex. By Lemma 4.14, the set R can b e decomp osed into the union R = S ∪ Y ∪ { x } suc h that S and Y satisfy Conditions (bi) – (biv) of Lemma 4.14. By Conditions (bi) , (bii) , and (biv) , S ⊆ N 2 ( x ) \ N ( Y ) m inimally dominates N ( x ) \ Y , and thus Item (b) holds. By Condition (biii) , each v ertex in Y has a priv ate neighbor with resp ect to Y ∪ { x } , and w e conclude Item (a) holds, hence completing the ﬁrst direction of the pro of. Assume that b oth conditions (a) and (b) hold. Item (b) establishes that N 2 ( x ) \ N ( Y ) dominates N ( x ) \ Y . Consequently , there exists an inclusion-wise minimal set D ⊆ N 2 ( x ) \ N ( Y ) that dominates N ( x ) \ Y . Let R = Y ∪ D ∪ { x } . Notice that D ∩ N ( Y ) is empty; otherwise, there would exist a vertex y ∈ Y suc h that its remov al from R w ould maintain x redundan t by our assumption on the cardinality of Y . Therefore, since | Y | ≥ 2 and D ∩ N ( Y ) = ∅ , we obtain by Lemma 4.14 that the set R is a minimal redundant set that satisﬁes the desired prop erties. This concludes our pro of. Lemma 4.18. L et x ∈ V ( G ) b e a vertex. If Y ⋆ ⊆ N ( x ) is extendable with r esp e ct to x , then every subset Y ⊊ Y ⋆ satisfying | Y | ≥ 2 is also extendable with r esp e ct to x . 21 Pr o of. Let Y ⋆ b e an extendable set and Y ⊊ Y ⋆ b e a proper non-empt y subset of Y ⋆ . By Lemma 4.17, each v ertex y ∈ Y ⋆ has a priv ate neighbor z y ∈ N ( y ) . Consequently , the v ertices in Y ⋆ \ Y can b e dominated by these priv ate neigh b ors, hence b y N 2 ( x ) \ N ( Y ) . By Lemma 4.17 since | Y | ≥ 2 w e conclude that Y is an extendable set. Let Y b e an extendable set. F ollowing Lemmas 4.14 and 4.17, the minimal redundant sets R suc h that red ( R ) = { x } and R ∩ N ( x ) = Y are precisely those of the form S ∪ Y ∪ { x } where Y is extendable and S ⊆ N 2 ( x ) \ N ( Y ) minimally dominates N 2 ( x ) \ N ( Y ) . Again, this can be framed as red-blue domination: w e need to minimally dominate N ( x ) \ Y (blue) with vertices from N 2 ( x ) \ N ( Y ) (red). Therefore, the algorithm for this part lists all extendable sets Y using an algorithm to en umerate all members of an indep endence set system, discarding singletons and the empty set, and for each such Y solv es the corresp onding instance of red-blue domination. Note that, as long as the algorithm for red-blue domination do es not pro duce rep etitions, this algorithm will not either, as extendable Y ’s partition the solutions to b e enumerated. As for complexity , observe ﬁrst that whether a set is extendable can be tested in p olynomial time. Giv en that the members of an indep endence set system can b e en umerated with p olynomial delay provided they can b e recognized in p olynomial time (see, e.g., [ KLMN14 ] for an example), we derive the follo wing prop ositions, mimic king Prop ositions 4.15 and 4.16: Prop osition 4.19. L et G b e a ( C 3 , C 5 , C 6 ) -fr e e gr aph and let x ∈ V ( G ) . Ther e is an incr emental-quasi-p olynomial time algorithm that lists al l minimal r e dundant sets R wher e red ( R ) = { x } and R ∩ N ( x ) = Y for some Y ⊆ N ( x ) and | Y | ≥ 2 . Prop osition 4.20. L et G b e a ( C 3 , C 5 , C 6 , C 8 ) -fr e e gr aph and let x ∈ V ( G ) . Ther e is a p olynomial-delay algorithm that lists al l minimal r e dundant sets R wher e red ( R ) = { x } and R ∩ N ( x ) = Y for some Y ⊆ N ( x ) and | Y | ≥ 2 . Com bining the ab o ve prop ositions w e obtain Theorem 1.7 that we restate b elow. Let us observe that a solution of the form N [ x ] might b e obtained twice: at prepro cessing or in the algorithm used for ﬁnding all minimal redundan t sets where x is the unique redundan t vertex. Since the rep etition is unique, this, ho wev er, can b e handled without impact on the delay by simply not outputting the solution the second time it is pro duced. Theorem 1.7. Ther e is a p olynomial-delay algorithm that solves GraphMred · Enum on ( C 3 , C 5 , C 6 , C 8 ) -fr e e gr aphs, and an incr emental quasi-p olynomial-time algorithm that solves GraphMred · Enum on ( C 3 , C 5 , C 6 ) -fr e e gr aphs. 4.2 Co-bipartite graphs In this section, we prov e that, unless P = NP , there is no output-p olynomial time algorithm solving GraphMred · Enum in co-bipartite graphs. Our reduction relies on earlier results of Boros and Makino [ BM24 ], stating that HypMred · Enum is intractable for h yp ergraphs of dimension 3. Nevertheless, the problem can b e solved in p olynomial time for h yp ergraphs of maximum degree 3 due to every solution having size at most 4. W e consider a h yp ergraph H of dimension 3, whose transp osed h yp ergraph H t has maxim um degree at most 3. By lo oking at the co-bipartite incidence graph C ( H ) of H , w e prov e that listing the members of MinRed ( C ( H )) amounts to en umerating MinRed ( H ) , 22 together with a p olynomial num b er of solutions of size at most 4—including MinRed ( H t ) and solutions spread across the parts of C ( H ) . Theorem 4.21. The pr oblem GraphMred · Enum r estricte d to c o-bip artite gr aphs c annot b e solve d in output-p olynomial time unless P = NP . Pr o of. Let H b e a hypergraph of dimension 3 and consider its co-bipartite incidence graph C : = C ( H ) . Recall that the v ertex set of C is partitioned into sets V and U corresp onding to the v ertices and edges of H , respectively . If e is an element of U b y E w e mean its corresp onding hyperedge. W e ﬁrst relate the minimal redundant sets of C ( H ) with the minimal redundant sets of H . Namely , we show the following equality: MinRed ( C ) = X ∪ R with X = { R | R ∈ MinRed ( C ) , | R | ≤ 4 } R = { R | R ∈ MinRed ( H ) , | R | ≥ 5 } W e start by pro ving the inclusion MinRed ( C ) ⊆ X ∪ R . Let R ∈ MinRed ( C ) . Let us further assume that | R | ≥ 5 , as the inclusion trivially holds for smaller R . Note that for an y v ∈ V , e ∈ U , v e is dominating C . Therefore, v , e ∈ R w ould imply that | R | ≤ 3 . Hence, either R ⊆ U or R ⊆ V . W e prov e that R ⊆ V . Given that | R | > 4 and U, V are cliques, no v ertex of R is self-priv ate. Now, since H t is of degree at most 3, each vertex in U has at most 3 neighbors in V . Hence to mak e a given e ∈ U redundan t, that is to dominate N [ e ] , at most 3 v ertices are needed. It follows that a minimal redundant set of C included in U is of size at most 4. W e deduce that R ⊆ V as exp ected. No w since V is a cli que and | R | ≥ 5 , no vertex x in R gains a priv ate neighbor in V when removing another vertex from R . Consequen tly R is minimal with the prop erty that not all its elemen ts hav e priv ate neighbors in U . As v e is an edge of C if and only if v ∈ E in H , we derive that R is a minimal redundan t set of H . W e now turn to the other inclusion. Let R ∈ MinRed ( H ) with | R | ≥ 5 . Then R ⊆ V . Again, note that no element of V can b e a priv ate neigh b or of a vertex in R , nor of a v ertex in R \ { x } for an y x ∈ R . As v e is an edge of C if and only if v ∈ E in H , w e deriv e that R is a minimal redundant set of C . W e are ready to conclude. W e show that an output-p olynomial time algorithm A for solving GraphMred · Enum in co-bipartite graphs can b e used to pro duce an output- p olynomial time algorithm B solving HypMred · Enum in h yp ergraph of dimension 3. This would entail P = NP due to [ BM24 ]. The algorithm B ﬁrst runs ov er all subsets of V of size ≤ 4 and identiﬁes among them the minimal redundant sets of H with size ≤ 4 . Then, it builds the graph C in p olynomial time and runs the algorithm A on input C . Based on the previous equality , A will indeed pro duce all remaining minimal redundan t sets of H . Giv en that C is of p olynomial size in the size of H , and | MinRed ( C ) | ≤ | MinRed ( H ) | + ( | V ( H ) + | E ( H ) | ) 4 , the algorithm A will run in a time polynomially b ounded by the sizes of | H | and | MinRed ( H ) | . This completes the description of B , whic h also runs in output-p olynomial time as exp ected, which concludes the pro of. Note that our reduction preserves the delay as there is only a p olynomial num b er of solutions to discard. 23 4.3 Split graphs In this section, we sho w that GraphMred · Enum remains intractable ev en when restricted to split graphs. This result contrasts with GraphMirr · Enum , as the latter can b e solv ed with p olynomial delay and space; see Theorem 1.3. The reduction w e use is the split incidence graph S 2 ( H ) of an arbitrary hypergraph H . Recall that S 2 ( H ) is obtained from B ( H ) by completing V in to a clique. W e then argue that en umerating the minimal redundant sets of S 2 ( H ) amount to list the minimal redundant sets of H plus a p olynomial n umber of solutions of size 2 , following similar arguments as in the pro of of Theorem 4.21. Theorem 4.22. The pr oblem GraphMred · Enum r estricte d to split gr aphs c annot b e solve d in output-p olynomial time unless P = NP . Pr o of. Let H b e a hypergraph with vertices V = { v 1 , . . . , v n } and edges E = { E 1 , . . . , E m } , and consider the split incidence graph S : = S 2 ( H ) . Recall that its vertex set is partitioned in to V and U where V induces a clique and U = { e 1 , . . . , e m } induces an indep endent set where e j represen ts E j . W e ﬁrst characterize MinRed ( S ) in terms of MinRed ( H ) : MinRed ( S ) = X ∪ R with X = { R | R ∈ MinRed ( S ) , | R | = 2 } R = { R | R ∈ MinRed ( H ) , | R | ≥ 3 } W e ﬁrst prov e the inclusion MinRed ( S ) ⊆ X ∪ R . Let R ∈ MinRed ( S ) . Let us further assume that | R | ≥ 3 , as the inclusion trivially holds for | R | = 2 , and no minimal redundan t set has less than tw o elements. Observ e that for any v ∈ V and an y e ∈ U , if v is adjacent to e then v e is a minimal redundant set of size 2 as N [ e ] ⊆ N [ v ] . Th us R do es not contain any suc h edge. Let v b e a redundant vertex of R , which, b y the previous argumen t, must thus lie in V . As N [ v ] is dominated b y R \ { v } , but R ∩ U ∩ N ( v ) = ∅ , there must exist w ∈ V ∩ R distinct from v that preven ts v from b eing self-priv ate. Supp ose tow ard a contradiction that there exists e ∈ U ∩ R . Since R ∩ U ∩ N ( v ) = ∅ , N [ e ] ∩ N [ v ] ⊆ N [ w ] ∩ N [ v ] and so N [ v ] remains dominated by R \ { e } , contradicting the minimalit y of R . So R ⊆ V . No w since V is a cli que and | R | ≥ 3 , no vertex x in R gains a priv ate neighbor in V when removing another vertex from R . Consequen tly R is minimal with the prop erty that not all its elements ha ve priv ate neighbors in U . As v e is an edge of S if and only if v ∈ E in H , we derive that R is a minimal redundan t set of H . W e now turn to the other inclusion. Let R ∈ MinRed ( H ) with | R | ≥ 3 . Then R ⊆ V . Again, note that no element of V can b e a priv ate neigh b or of a vertex in R , nor of a v ertex in R \ { x } for an y x ∈ R . As v e is an edge of S if and only if v ∈ E in H , we deriv e that R is a minimal redundant set of S . W e are ready to conclude. W e prov e that an output-p olynomial tim e algorithm A for solving GraphMred · Enum in split graphs yields an output-p olynomial time algorithm B solving HypMred · Enum . This en tails P = NP due to [ BM24 ]. The algorithm B ﬁrst identiﬁes the minimal redundant sets of H of size at most 2 by ranging ov er singleton and pairs of vertices of V . Then, it builds the graph S in p olynomial time and runs the algorithm A on input S . Based on the previous equality , A will indeed pro duce all remaining minimal redundan t sets of H in time | S | + | MinRed ( S ) | . Since 24 v 4 G E 1 E 2 E 3 E 4 v 1 v 2 v 3 v 4 e 1 e 2 y z G e 3 e 4 N [ v 4 ] ∈ H N [ e 2 ] ∈ H := N ( G ) v 3 v 1 v 2 Figure 6: The reduction of Theorem 5.2. W e start from a hypergraph G (on the left), build its bipartite incidence graph B ( G ) to which we add a vertex y univ ersal to V and a v ertex z p ending at y . The resulting graph G is pictured on the right. Then, w e consider the closed neighborho o d hypergraph H : = N ( G ) . T w o of its edges, N [ e 2 ] and N [ v 4 ] , are dra wn on G . | S | + | MinRed ( S ) | ≤ | MinRed ( H ) | + | V | 2 , we obtain that B runs in output-p olynomial time as exp ected, whic h concludes the pro of. 5 P ersp ectiv es In this pap er, we studied the problems GraphMirr · Enum and GraphMred · Enum in graph classes capturing incidence relations suc h as co-bipartite, split and bipartite graphs. In particular, we prov ed the hardness of GraphMred · Enum on split and co-bipartite graphs, while for GraphMirr · Enum w e show ed that the problem is as hard in general graphs as in co-bipartite graphs. Question 1.2 remains thus op en for GraphMirr · Enum . Concerning tractabilit y , in addition to the case of GraphMirr · Enum admitting a p olynomial-delay algorithm on split graphs, originally claimed by Uno in [ BBHK15 ], w e show ed that b oth problems admit p olynomial-delay algorithms on generalizations of c hordal bipartite graphs. The case of bipartite graphs is left op en, which leads to the follo wing question: Question 5.1. What is the c omplexity of GraphMirr · Enum and GraphMred · Enum in bip artite gr aphs? T ow ards this question, we conclu de this pap er by giving evidence that the algorithms w e prop osed do not straightforw ardly apply to the bipartite case. First, we pro ve that the sequen tial metho d we applied to solve GraphMirr · Enum in strongly orderable graphs cannot b e used on bipartite graphs. Theorem 5.2. Unless P = NP , ther e is no output-p olynomial time algorithm which, given a bip artite gr aph G , an or dering v 1 , . . . , v n of its vertic es, an inte ger i ∈ [ n − 1] , and I ⋆ ∈ MaxIrr ( H i ) wher e H = N ( G ) , enumer ates children ( I ⋆ , i ) . 25 Pr o of. Let G b e a instance of HypMirr · Enum in hypergraphs, for which the existence of an output-p olynomial time algorithm implies P = NP [ BM24 , Theorem 2]. Let us assume without loss of generality that ev ery v ertex of G app ears in at least one hyperedge, that G do es not contain the empty h yp eredge, and let us denote b y N and M the num b er of v ertices and edges of G , resp ectively . Let us ﬁnally assume that G is not a trivial instance in whic h V ( G ) is irredundant. No w, let G b e the bipartite incidence graph B ( G ) of G extended with a vertex y adjacen t to ev ery v ertex in V = V ( G ) , and a vertex z adjacen t only to y . Lastly , we consider the ordering e 1 , . . . , e M , v 1 , . . . , v N , y , z of the vertices of G , and set H : = N ( G ) , I ⋆ : = V , and i : = N + M . This concludes the construction of the instance. It is illustrated on an example in Figure 6. The set V is a maximal irredundant set of H i , b ecause ev ery v ertex in V is self-priv ate and every other vertex in V i is dominated. Notice that by our assumptions V ∪ { y } is redundan t, and that y has a priv ate hyperedge in H i +1 , namely the trace { y } of N [ z ] . Since the vertices in V cannot b e self-priv ate, the extensions of V are of the form Y ∪ { y } where Y is a maximal subset of V ha ving priv ate neighbors in E . Note that the children of I ⋆ = V with resp ect to i are precisely the family of all suc h extensions. Moreo ver, notice that suc h se ts Y are the maximal irredundant sets of G ; otherwise, we can add another vertex from V \ Y in to Y , contradicting its maximality . As the construction can b e conducted in p olynomial time, we get the desired result. As for GraphMred · Enum , the strategy w e used in graphs without small cycles relied on tw o crucial facts: (1) a minimal redundan t set has a constan t num b er of redundan t v ertices (at most 3 ), and (2) the minimal redundant sets containing a single redundant v ertex and a unique neighbor of this vertex can b e enumerated eﬃciently . In bipartite graphs these t wo facts do not hold anymore, as we show now. F or redundant sets with un b ounded n umber of redundant vertices, let us consider the follo wing bipartite graph G . W e start from vertices v 1 , . . . , v n and create, for eac h i  = j a new vertex u ij adjacen t to v i and v j . Then, we add a vertex y adjacen t to eac h v i and a v ertex z adjacen t to y . This complete the description of G . The set R : = { y , v 1 , . . . , v n } is a minimal redundant set of G where red ( R ) = { v 1 , . . . , v n } is indeed of unbounded size. W e illustrate this construction in Figure 7. F or the task of enumerating the minimal redundant sets with a single redundant v ertex x and con taining a unique neigh b or y of x , i.e., the case of Lemma 4.13, we show in the next theorem that it b ecomes intractable in bipartite graphs. The hardness comes from the fact that, when selecting a subset of N ( y ) to b e part of the redundant set, one m ust guarantee that each chosen v ertex has a priv ate neighbor outside N 2 ( x ) all the while b eing useful to dominate N ( x ) \ { y } , a problem that does not arise in bipartite graphs with no induced C 6 . Theorem 5.3. L et G b e a bip artite gr aph and x, y ∈ V ( G ) b e a p air of adjac ent vertic es. Then, unless P = NP , ther e is no output-p olynomial time algorithm to enumer ate the minimal r e dundant sets R of G that satisfy red ( R ) = { x } and R ∩ N ( x ) = { y } . Pr o of. Let ϕ b e an instance of 3 -SA T with literals X = { x 1 , x 1 , . . . , x n , x n } and clauses C : = { c 1 , . . . c m } . W e construct the incidence bipartite graph G of ϕ with partition ( X, C ) . Moreo ver, we add t wo adjacent vertices x and y , with x complete to C , y complete to X , and a vertex u whose only neighbor is y . Finally , for each i ∈ [ n ] , we create a new v ertex 26 y red ( R ) v 1 v 2 v 3 v 4 u 12 u 13 u 14 u 23 u 24 u 34 z Figure 7: A bipartite graph with a minimal redundant set (shaded vertices) having an un b ounded num b er of redundant v ertices (b oxed in yello w). The vertex z sho ws that a minimal redundan t set with an un b ounded num b er of redundant vertices do es not need to b e the closed-neigh b orho o d of a vertex. y i adjacen t to both x i and x i . This ﬁnishes our construction of G . An illustration of the construction is depicted in Figure 8. W e ﬁrst pro ve that there exists a minimal redundant set R of G suc h that red ( R ) = { x } and R ∩ N ( y ) = { x } if and only if ϕ has a satisfying truth assignment. W e start with the if part. Let A b e the set of literals set to true in a minimal partial satisfying assignmen t of ϕ and let R := A ∪ { x, y } . W e show that R satisﬁes the aforementioned requirements. W e readily hav e that R ∩ N ( x ) = { y } b y construction of G . Since A is a partial satisfying assignmen t of ϕ , eac h vertex in C is dominated b y a vertex of A . Therefore, R indeed dominates N [ x ] , making x redundan t. In R , y has priv ate neighbor u and an y literal ℓ i has priv ate neigh b or y i as R ⊆ X ∪ { x } and A cannot contain b oth x i , ¯ x i for any i ∈ [ n ] b y assumption. W e thus obtain that red ( R ) = { x } . T o see that R is minimal, observe that discarding y w ould make x self-priv ate and that discarding any literal of A w ould giv e a priv ate neigh b or c j to x b y assumption on the minimality of A for satisfying ϕ . As all other vertices of R are irredundant, we deduce that R is indeed minimally redundant. c 1 c 2 c 3 x 1 ¯ x 1 x 2 ¯ x 3 y 1 y 2 y 3 u y x 3 ¯ x 2 N ( x ) \ { y } N ( y ) \ { u } x Figure 8: The reduction of Theorem 5.3. Blue zones indicate the vertices representing clauses of ϕ , while the red zones represent the literals. The shaded vertices form a minimal redundan t set where the (y ellow) b old vertex is redundant. Those in N ( y ) \ { u } corresp ond to a partial satisfying assignment of ϕ . 27 W e turn to the only if part. Let R b e a minimal redundan t set of G suc h that red ( R ) = { x } and R ∩ N ( x ) = { y } . Let A := R \ { x, y } . By assumption, N ( x ) \ { y } is minimally dominated b y A , and by construction it must thus b e that A ⊆ X . As red ( R ) = { x } , X ⊆ N ( y ) , and N ( ℓ i ) \ { y i } ⊆ N ( x ) for any literal ℓ i , an y vertex ℓ i m ust ha ve y i as priv ate neighbor. W e deduce that at most of one x i , ¯ x i b elongs to A for all i ∈ [ n ] . This mak es A corresp ond to a partial assignment of the v ariables x 1 , . . . , x n where the literals in A are set to true. Since N ( x ) \ { y } is dominated by A , we deduce that eac h clause in ϕ con tains a literal in A . Thus, A corresp onds to a partial satisfying assignmen t of ϕ as desired. Let us now call R the set of minimal redundant sets R of G that satisfy red ( R ) = { x } and R ∩ N ( x ) = { y } and assume that there exists an output-p olynomial time algorithm A listing the members of R in time ( | G | + | R | ) c for some constant c . W e show that A can b e used to decide whether ϕ is satisﬁable in p olynomial time. W e build G in time p olynomial in the size of ϕ then run A with input G for ( | G | + 1) c time. 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Generating minimal redundant and maximal irredundant sets in incidence graphs

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