Extremal graphs for the identifying code problem

Extremal graphs for the iden tifying co de problem ✩ Florent F oucaud a , Eleonora Guer rini b , Matjaž Kovše a , Reza Naserasr a , Aline P arreau b , Petru V a licov a a L aBRI - Université Bo r de aux 1 - CNRS, 351 c ours de la Lib ér ation, 33405 T alenc e c e dex, F r anc e. b Institut F ourier 100, rue des Maths, BP 74, 38402 St Martin d’Hèr es Ce dex, F r anc e Abstract An identifying co de of a gr aph G is a domina ting set C such that every v ertex x of G is distinguished from other vertices b y the set of vertices in C that are at distance at most 1 fro m x . The problem of finding an identif ying co de of minimum pos s ible size turned o ut to b e a challenging problem. It was prov ed by N. Bertra nd, I. Charo n, O. Hudry and A. Lobstein tha t if a g raph on n vertices with at least one edg e admits an iden tifying code, then a minimal iden tifying code has size at most n − 1 . They intro duced classes of graphs whose smallest iden tifying c o de is of size n − 1 . F ew conjectures w ere for m ula ted to classify the class o f a ll graphs whose minim um identifying co de is of size n − 1 . In this pap e r , disproving these conjectures, we classify all finite gr aphs for which a ll but one of the vertices are needed to for m an identifying co de. W e classify a ll infinite gr aphs needing the whole set o f vertices in any iden tifying co de. New upper bounds in terms of the num b er of vertices and the maxim um degree o f a gra ph are also pr o vided. Keywor ds: Iden tifying co des, Dominating sets, Infinite graphs. 1. In tro duction Given a g raph G , an ident ifying co de of G is a subset C o f vertices of G such that the subset of C at distance at most 1 from a g iv e n v ertex x is no nempt y a nd uniquely determines x . Iden tifying co des hav e bee n widely studied since the intro duction of the concept in [14], and have b een a pplied to pr oblems such as fault-diagnosis in multiprocess o r systems [14], compact routing in netw orks [15], emerge ncy sensor net works in facilities [17] or the analysis of secondary RNA s tructures [13]. The concept of iden tifying co des of graphs is rela ted to s ev eral other concepts, such as lo cating-dominating sets [2 0, 19] for graphs and the well-celebrated theor em of Bondy [1] on set systems. The purpos e of this pa per is to classify extremal cases in some previously known upp er b ounds for the minim um size of identifying co des and thus a ls o improving tho se upp er bounds. W e b egin b y in tro ducing our terminolo gy . Unless sp ecifically mentioned G = ( V , E ) will b e a finite simple graph with n = | V | being the num b er o f vertices. The degree o f a vertex x is denoted deg ( x ) . By ∆( G ) we denote the maximum degree of G . F or tw o vertices x and y o f G , w e denote by d G ( x, y ) (or d ( x, y ) if there is no ambiguit y) the distance betw een x and y in G . The b al l of ra dius r centered at x , denoted B r ( x ) , is the set of vertices at distance at most r of x . W e note that x b elongs to B r ( x ) for every r . A vertex x o f G is universal if B 1 ( x ) = V ( G ) . Given a subset S of V ( G ) , we say that a vertex x is S -universal if S ⊆ B 1 ( x ) . The symmetric difference of tw o sets A and B is denoted by A ⊖ B . Given a pair of vertices of a gr aph G , we write ⊖ r ( x, y ) = B r ( x ) ⊖ B r ( y ) . T wo vertices x and y are ca lled twins in G if B 1 ( x ) = B 1 ( y ) . A g raph is called twin-fr e e if it has no pair o f t win vertices. The complemen t of a graph G is denoted by G . F or r ≥ 2 , the r th -p ower o f G , is the gra ph G r = ( V , E ′ ) with E ′ = { xy | x, y ∈ V , d G ( x, y ) ≤ r } . Conv er sely if H r ∼ = G , then w e say H is an r -r o ot ✩ This researc h is supported by the ANR Pro ject IDEA • ANR-08-EMER-007, 2009-2011. Pr eprint submitte d to Eur op e an J. Combin. Novemb er 10, 2021 of G . W e denote by G − x the gra ph obtained fro m G b y removing x fro m V ( G ) a nd all edge s containing x fro m E ( G ) . F or t w o g raphs G 1 = ( V 1 , E 1 ) a nd G 2 = ( V 2 , E 2 ) , G 1 ⊲ ⊳ G 2 is the join gr aph of G 1 and G 2 . Its v ertex set is V 1 ∪ V 2 and its edge set is E 1 ∪ E 2 ∪ { x 1 x 2 | x 1 ∈ V 1 , x 2 ∈ V 2 } . W e denote by K n , the complete gr aph on n vertices, by P n , the path on n v ertices, a nd by K a,b , the complete bipartite graph with bipartitions of sizes a and b . Given a gr aph G and an integer k ≥ 2 , a subset I of vertices of G is ca lled a k -indep en dent set if for all distinct vertices x, y of I , d G ( x, y ) ≥ k . A 2 -indep enden t set is simply an indep endent set . Given an in teger r ≥ 1 , a subset S of vertices o f G is called an r -dominating set if for every vertex x o f G , B r ( x ) ∩ S 6 = ∅ . W e say that S r - sep ar ates t wo vertices x and y , if B r ( x ) ∩ S 6 = B r ( y ) ∩ S . A subset S of v ertices is an r -sep ar ating set if it r -separa tes all distinct vertices x, y of G . If S is bo th r -do minating and r -separating , S is an r -identifying c o de [14]. If S is r -dominating and r -separates v ertices of V ( G ) \ S , it is called an r -lo c ating-dominating set [20]. Given a bipar tite graph G with a partition V = I ∪ A , a subset S of A is said to b e an r -discriminating c o de [5] if S r -separates a ll pairs of distinct vertices of I . In each of the prev ious concepts when r = 1 , we simply use the name of the concept without sp ecifying the v alue o f r . Note that a set C is an r -separating set o f G (resp. r -identifying co de) if and only if it is a separa ting set (resp. identifying co de) of G r . A graph G admits a separa ting set (re s p. identifying co de) if and only if it is t win-free, a s a conse q uence it admits an r -separ ating set (res p. r -identif ying co de) if a nd o nly if G r is t win-free [6]. F or a gra ph G , the minim um cardinalities o f an r -dominating set and of an r -lo cating-do minating set are commonly deno ted by γ r ( G ) and γ LD r ( G ) . If G r is twin-free, we denote by γ ID r ( G ) (resp ectively γ S r ( G ) ) the minimum cardinality of a n r - iden tifying co de ( r -separating set) o f G . It is clear from the definition that γ S r ( G ) ≤ γ ID r ( G ) ≤ γ S r ( G ) + 1 . While the exact v a lue of γ ID for so me cla sses of g raphs has b een deter mined [3, 4], finding the v alue of γ ID r ( G ) for a gener al graph G is known to b e NP-hard for any r ≥ 1 [9 , 7]. Upper bo unds, in terms of basic graph parameters , have b een g iv e n for the minimum sizes of the corresp onding sets for mo s t o f the previously defined concepts. In pa rticular it has been s hown that γ LD r ( G ) ≤ | V ( G ) | − 1 and, assuming G is t win-free and G 6 ∼ = K n , γ ID r ( G ) ≤ | V ( G ) | − 1 (see [19, 12, 8]). F or the case of lo cating-dominating sets, it was prov ed in [1 9] that for a connected graph G we hav e γ LD ( G ) = | V ( G ) | − 1 if and only if G is either a star or a complete graph. In this pap er, we do the analog o us c la ssification for iden tifying co des. In the case of iden tifying co des, the class of graphs re a c hing this b ound is a muc h richer family . Thus w e answer, in negative, the tw o attempted conjectures for such classificatio n [1 8, 5]. This gives a partial a nsw er to a q ues tion p o sed in [5]. This is done in Section 3. All the previous definitions ca n easily b e extended to infinite graphs. Exa mples of nontrivial infinite graphs fo r which the whole v ertex set is needed to form an identif ying co de a r e given in [8]. W e class ify a ll such infinite gra phs in Section 4 . In Section 5 we in tro duce new upp er b ounds for γ ID in terms of n and ∆ . In a ll these sections w e addres s the problem o f iden tifying co des only for r = 1 . In Section 6 we consider general r -identif ying co des. The next section provides a set of preliminary results. 2. Preliminary results In this section w e hav e put tog ether so me basic results necessa ry for our ma in work. These res ults could be us eful in the study of iden tifying co des in general. W e sta rt by recalling the following theo r em. Theorem 1 ([2, 12]) . L et G b e a twin-fr e e gr aph on n vertic es having at le ast one e dge. Then γ ID ( G ) ≤ n − 1 . It is shown in [8] that this b ound is tight. In particular it is shown that for a n y t ≥ 2 , γ ID ( K 1 ,t ) = t . A stronger re s ult is prov ed in Section 5 (see Lemma 19). The next lemma is an obvious but a crucial one. 2 Lemma 2. L et G b e a twin-fr e e gr aph and let C b e an identifying c o de of G . Then, any set C ′ ⊆ V ( G ) such that C ⊆ C ′ is an identifying c o de of G . The next prop osition is useful in proving upper b ounds on minimum identif ying co des by induction. Prop osition 3. L et G b e a twin-fr e e gr aph and S ⊆ V ( G ) su ch t hat G − S is twin-fr e e. Then γ ID ( G ) ≤ γ ID ( G − S ) + | S | . Pr o of. T ake a minimum co de C 0 of G − S . Consider the vertices of S in an a r bitrary order ( x 1 , . . . , x | S | ) . Using induction w e extend C 0 to a subset C i of G whic h identifies the vertices in V i = V ( G ) \ { x i +1 , . . . , x | S | } . T o do this, if C i − 1 iden tifies a ll the vertices of V i , we are done. Other wise, since all the vertices in V i − 1 are iden tified, either B 1 ( x i ) ∩ C i − 1 = B 1 ( y ) ∩ C i − 1 for exactly one vertex y in V i − 1 , or x i is not dominated by C i − 1 . In the first case x i and y are separated in G by some v ertex, say u , so let C i = C i − 1 ∪ { u } . In the second case, let C i = C i − 1 ∪ { x i } . Now, in b oth ca s es, C i iden tifies all the vertices of V i . A t step | S | , C | S | is a n ident ifying co de of G of size at most | C 0 | + | S | ≤ γ ID ( G − S ) + | S | . W e will need the following sp ecial case of the pr e v ious prop osition. Corollary 4. L et G b e a c onne cte d gr aph with γ ID ( G ) = | V ( G ) | − 1 , G ≇ K 1 , 2 , then t her e is a vertex x of G su ch t hat G − x is stil l c onne cte d and γ ID ( G − x ) = | V ( G − x ) | − 1 . Pr o of. If G ∼ = K 1 ,t , t 6 = 2 , then any leaf vertex works. Thus, w e ma y suppose G ≇ K 1 ,t . Then by Theo rem 1, there is a vertex x of G such that V ( G − x ) is an identifying co de of G and th us G − x is t win- free and G − x ≇ K n . By Prop osition 3, we have γ ID ( G − x ) ≥ γ ID ( G ) − 1 = | V ( G − x ) | − 1 . Equalit y ho lds since other wise γ ID ( G ) = | V ( G ) | . T o complete the pro of, w e show that x can b e chosen such that G − x is connected. T o see this, assume G − x is not connected. Since γ ID ( G − x ) = | V ( G − x ) | − 1 , except one comp onen t, every compo nen t of G − x is a n is olated vertex. If there a r e tw o or more such isolated vertices, then either one of them can be the vertex we want. Otherwise the r e is only one isolated vertex, call it y . Now if G − y is twin-free, then y is the desire d vertex, else ther e is a vertex x ′ such that B 1 ( x ′ ) = B 1 ( x ) − y . Then G − x ′ is connected and t win-free. Lemma 5. L et G b e a twin-fr e e gr aph and let v ∈ V ( G ) . L et x, y b e a p air of twins in G − v . If G − x or G − y has a p air of t wins, then v must b e one of the vertic es of the p air. Pr o of. Since v separates x and y , it is adjacent to one of them (say x ) and not to the other. Supp ose z , t ar e t wins in G − x . Supp ose z is a djacen t to x and t is not. If z 6 = v then y is also adjacent to z and, therefore , t is also adjacent to y which implies x be ing adjace nt to t . This contradicts the fact that x s e parates z and t . The other case is proved simila rly . Prop osition 6. L et G 1 and G 2 b e twin-fr e e gr aphs such that for every minimum sep ar ating set S ther e is an S -u niversal vertex . If G 1 ⊲ ⊳ G 2 is twin-fr e e, then we have γ S ( G 1 ⊲ ⊳ G 2 ) = γ S ( G 1 ) + γ S ( G 2 ) + 1 . F urthermor e, if S is a sep ar ating set of size γ S ( G 1 ) + γ S ( G 2 ) + 1 of G 1 ⊲ ⊳ G 2 , then t her e is an S -u niversal vertex. Pr o of. Let S b e a minimum sepa rating s et of G 1 ⊲ ⊳ G 2 . Since vertices of G 2 do not separate any pair of vertices in G 1 then S ∩ V ( G 1 ) is a separating set of G 1 . B y the same a rgument S ∩ V ( G 2 ) is a separating set of G 2 . Therefore, | S | ≥ γ S ( G 1 ) + γ S ( G 2 ) . But if | S | = γ S ( G 1 ) + γ S ( G 2 ) , then ther e is an [ S ∩ V ( G 1 )] -universal vertex x in G 1 and an [ S ∩ V ( G 2 )] -universal vertex y in G 2 . But then, x and y are not separ ated b y S . Given a sepa r ating set S 1 of G 1 and a separating set S 2 of G 2 , the set S 1 ∪ S 2 separates all pairs of v ertices except the S 1 -universal vertex of G 1 from the S 2 -universal vertex of G 2 . B ut since G 1 ⊲ ⊳ G 2 is t win-free, we could a dd one more vertex to S 1 ∪ S 2 to obtain a se parating set of G 1 ⊲ ⊳ G 2 of size γ S ( G 1 ) + γ S ( G 2 ) + 1 . F or the se c o nd part assume S is a separating set o f s iz e γ S ( G 1 ) + γ S ( G 2 ) + 1 of G 1 ⊲ ⊳ G 2 . T hen we hav e either | S ∩ V ( G 1 ) | = γ S ( G 1 ) or | S ∩ V ( G 2 ) | = γ S ( G 2 ) . Without loss of g eneralit y assume the fo rmer. Then there is an [ S ∩ V ( G 1 )] -universal vertex z of G 1 . Since z is also adjacent to all the vertices of G 2 , it is an S - universal vertex of G 1 ⊲ ⊳ G 2 . 3 In Prop osition 6 if G 1 ≇ K 1 and G 2 ≇ K 1 , then γ ID ( G 1 ⊲ ⊳ G 2 ) = γ S ( G 1 ⊲ ⊳ G 2 ) = γ S ( G 1 ) + γ S ( G 2 ) + 1 . The following lemma was discov er ed in a discussion b et w een the first author, R. Klasing and A. Kosowski. W e include a pr oof for the sake of completeness . Lemma 7 ([11]) . L et G b e a c onne ct e d twin-fr e e gr aph, and I b e a 4 -indep endent set such that for every vertex x of I , the set V ( G ) \ { x } is an identifying c o de of G . Then C = V ( G ) \ I is an identifyi ng c o de of G . Pr o of. Clear ly C is a dominating se t of G . Let x, y b e a pair of v ertices o f G . If they b o th b elong to I , C ∩ B 1 ( x ) 6 = C ∩ B 1 ( y ) b ecause of the distance b etw een x and y . Otherwise, one o f them, say x , is in C . If they ar e not separa ted by C , then they must be adjacen t. Thus, together they co uld have only one neighbour in I , call it u . This is a contradiction b ecause V ( G ) \ { u } identifies G . W e note that 4 is the be st po ssible in the previous lemma. F o r example, let G = P 4 and as s ume x and y ar e the tw o ends of G . It is easy to chec k that V ( G ) \ { x } and V ( G ) \ { y } a re b oth iden tifying co des o f G but V ( G ) \ { x, y } is not. 3. Graphs with γ ID ( G ) = | V ( G ) | − 1 In this section we classify a ll g r aphs G for which γ ID ( G ) = | V ( G ) | − 1 . As a lready mentioned, stars are examples of such graphs. T o classify the r est we show that sp ecial powers o f paths are the basic ex a mples of such g r aphs. Then we show that any other example is mainly obta ined from the join of so me bas ic elemen ts. Definition 8. F or an inte ger k ≥ 1 , let A k = ( V k , E k ) b e the gr aph with vert ex set V k = { x 1 , . . . , x 2 k } and e dge set E k = { x i x j   | i − j | ≤ k − 1 } . x k +1 x k +2 x k +3 ... x 2 k − 1 x 2 k x 1 x 2 x 3 ... x k − 1 x k Clique on { x k +1 , ..., x 2 k } Clique on { x 1 , ..., x k } Figure 1: The graph A k whic h needs | V ( A k ) | − 1 v ertices for any iden tifying co de An illustration of graph A k is given in Figur e 1. W e no te that for k ≥ 2 we have A k = P k − 1 2 k and A 1 = K 2 . It is also ea sy to c heck that the o nly non trivial automorphism of A k is the mapping x i → x 2 k +1 − i . It is not ha r d to observe that A k is twin -free, ∆( A k ) = 2 k − 2 and that A k and A k are co nnected if k ≥ 2 . Prop osition 9. F or k ≥ 1 , we have: γ S ( A k ) = 2 k − 1 with B 1 ( x k ) and B 1 ( x k +1 ) b eing the only sep ar ating sets of size 2 k − 1 of A k . F urthermor e, if k ≥ 2 , γ ID ( A k ) = 2 k − 1 . Pr o of. Let S be a s eparating set of A k . F or i < k , we hav e ⊖ ( x i , x i +1 ) = { x i + k } a nd for k < i ≤ 2 k − 1 , we hav e ⊖ ( x i , x i +1 ) = { x i − k +1 } . Thus, { x 2 , . . . , x 2 k − 1 } ⊂ S . But to sepa rate x k and x k +1 , we m ust add x 1 or x 2 k . It is now easy to s ee that V k \ { x 1 } = B 1 ( x k +1 ) a nd V k \ { x 2 k } = B 1 ( x k ) , each is a separa ting set of size 2 k − 1 . If k ≥ 2 , then they b oth dominate A k and there fo re are also identif ying c odes. In the previous pro of in fact we hav e a lso proved that: 4 Corollary 10. F or k ≥ 1 every minimum sep ar ating set S of A k has a S -universal vertex. Let A b e the clos ur e of { A i | i = 1 , 2 , . . . } with resp ect to op eration ⊲ ⊳ . It is shown b elo w that elements of A ar e also extremal graphs with r espect to b oth se pa rating sets and identifying co des. Prop osition 11. F or every gr aph G ∈ A , we have γ S ( G ) = | V ( G ) | − 1 . F urthermor e, every minimum sep ar ating set S of G has an S -universal vertex. Pr o of. The prop osition is true for basic elements of A by Prop osition 9 and by Corollar y 10. F or a general element G = G 1 ⊲ ⊳ G 2 it is true by Prop osition 6 and by induction. Corollary 12. If G ∈ A and G ≇ A 1 , then γ ID ( G ) = | V ( G ) | − 1 . F urther examples of gra phs extremal with resp ect to separating sets and identif ying co des can be obta ined b y a dding a universal vertex to each of the gr aphs in A , as w e prov e below. Prop osition 13. F or every gr aph G in A ⊲ ⊳ K 1 we have γ ID ( G ) = γ S ( G ) = | V ( G ) | − 1 . Pr o of. Assume G = G 1 ⊲ ⊳ K 1 with G 1 ∈ A , and assume u is the vertex corre s ponding to K 1 . Supp ose S is a minim um separating set of G . W e first note that s ince S ∩ V ( G 1 ) is a separating set of G 1 , we hav e | S ∩ V ( G 1 ) | ≥ | V ( G 1 ) | − 1 . But if | S ∩ V ( G 1 ) | = | V ( G 1 ) | − 1 , then b y Propo sition 11, there is an [ S ∩ V ( G 1 )] -universal v ertex y of G 1 . Then y is not s e pa rated fro m x . Thus | S ∩ V ( G 1 ) | = | V ( G 1 ) | and therefore S = V ( G 1 ) . It is easy to chec k that S is also an ident ifying co de. It was proved in [8] that γ ID ( K n \ M ) = n − 1 where K n \ M is the co mplete gra ph minus a max ima l matchin g. W e no te that this graph, for ev en v alues of n , is the join o f n 2 disjoint copies of A 1 , thus it belo ngs to A . F or o dd v alues of n , it is built from the prev ious graph by adding a universal vertex. So far we hav e seen that γ ID ( G ) = | V ( G ) | − 1 for G ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) , G 6 ∼ = A 1 . W e also know that γ ID ( K n ) = n . More examples of gra phs with γ ID ( G ) = | V ( G ) | − 1 ca n b e obtained b y adding isolated vertices. In the next theor em w e show that for any o ther twin -free graph G we hav e γ ID ( G ) ≤ | V ( G ) | − 2 . Theorem 14. Given a c onne cte d gr aph G , we have γ ID ( G ) = | V ( G ) | − 1 if and only if G ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) and G 6 ∼ = A 1 . Pr o of. The “if ” part o f the theor em is alr e a dy prov ed. The pro of of the “only if ” part is base d on induction on the num b er o f vertices of G . F or gra phs on a t most 4 vertices this is ea sy to check. Assume the claim is true for graphs on at most n − 1 vertices and, b y cont radiction, let G b e a t win-free graph on n ≥ 5 vertices such that γ ID ( G ) = n − 1 and G / ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) . By Co r ollary 4 there is a vertex x ∈ V ( G ) such that G − x is connected and γ ID ( G − x ) = | V ( G − x ) | − 1 . By the induction h ypo thesis we have G − x ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) . Dep ending o n which one of these 3 sets G − x belo ngs to, we will hav e 3 ca ses. Case 1 , G − x ∈ { K 1 ,t | t ≥ 2 } . In this case w e consider a minimum iden tifying co de C o f G − x . If C do es no t alr e a dy identify x then either deg ( x ) ≤ 3 or deg ( x ) ≥ n − 2 . W e leav e it to the r e a der to chec k that in each of these cases, there is a n identifying co de of s ize n − 2 . Case 2 , G − x ∈ A . W e consider tw o subc a ses. Either G − x ∼ = A k for some k or G − x = G 1 ⊲ ⊳ G 2 , with G 1 , G 2 ∈ A . (1) G − x ∼ = A k , for some k ≥ 2 . If x is adjacent to all the vertices of G − x , then G ∈ A ⊲ ⊳ K 1 and we are done. Otherwise there is a pair o f co nsecutiv e vertices of A k , say x i and x i +1 , such that one is adjacent to x and the o ther is not. By the symmetry of A k we may assume i ≤ k . W e claim that C = V ( G ) \ { x 1 , x } or C ′ = V ( G ) \ { x 2 k , x } is a n identifying co de of G . This would contradict our assumption. W e first consider C a nd no te that C ∩ V ( A k ) is an iden tifying co de o f A k . If x is also separated fr o m a ll the vertices of G − x then we are done. Otherwis e there will be tw o po s sibilities. First w e co ns ider the po s sibilit y: x is not a djacen t to x i and adjacent to x i +1 . In this case each vertex x j , j > i + k , is sepa rated from x by x i +1 and each vertex x j , j < i + k , is sepa rated from x by x i . 5 Thu s x is not separa ted from x i + k . In the other p ossibility , x is adjacent to x i and not a djacen t to x i +1 . A similar argument implies that x is separ ated from every vertex but x 1 . In either o f these tw o po ssibilities, C ′ would be a n identif ying co de. (2) G − x ∼ = G 1 ⊲ ⊳ G 2 with G 1 , G 2 ∈ A . If x is a djacen t to all the vertices of G − x , then G ∈ A ⊲ ⊳ K 1 and we a r e do ne. Th us there is a vertex, say y , that is not adjac e n t to x . Without loss of gener alit y , we ca n assume y ∈ V ( G 1 ) . Let C 1 be an identif ying co de of size γ ID ( G 1 ) = | V ( G 1 ) | − 1 of G 1 which c o n ta ins y . The e xistence of such an identifying co de b ecomes apparent from the pro of of Prop osition 11. Then C = C 1 ∪ V ( G 2 ) is an identif ying co de of G 1 ⊲ ⊳ G 2 of size | V ( G 1 ⊲ ⊳ G 2 ) | − 1 = | V ( G ) | − 2 . Thu s C do es not separate a vertex of G 1 ⊲ ⊳ G 2 from x . Ca ll this vertex z . Since y ∈ C , z is no t adjacent to y , hence z ∈ V ( G 1 ) . Therefore, z is adjac e n t to all the vertices of G 2 . So x should also be adjacent to all the vertices of G 2 . T hus we hav e G = ( G 1 + x ) ⊲ ⊳ G 2 and any minim um identifying co de of G 1 + x together with all vertices o f G 2 would form an identifying co de of G . This proves that γ ID ( G 1 + x ) = | V ( G 1 + x ) | − 1 . Since G 1 + x has less vertices than G , by induction hypothesis, we hav e G 1 + x ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) and G 6 ∼ = A 1 . Since G 1 ∈ A , a nd since x is not a djac en t to a v ertex of G 1 , we should hav e G 1 + x ∈ A but all graphs in A hav e an even n um ber of vertices and this is no t p ossible. Case 3 , G − x ∈ A ⊲ ⊳ K 1 . Suppo se G − x ∼ = A i 1 ⊲ ⊳ A i 2 ⊲ ⊳ . . . ⊲ ⊳ A i j ⊲ ⊳ K 1 and let u b e the v ertex corresp onding to K 1 . If x is a lso adjacent to u , then u is a universal vertex of G and G − u is also twin-free. In this case we apply the induction on G − u : by Prop osition 3, γ ID ( G − u ) = | V ( G − u ) | − 1 and by induction hypothesis G − u ∈ { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) . But if G − u ∈ { K 1 ,t | t ≥ 2 } ∪ ( A ⊲ ⊳ K 1 ) , there will b e t wo univ ersal vertices, a nd therefore twins. Thus G − u ∈ A and G ∈ A ⊲ ⊳ K 1 . W e now assume x is not a djac e n t to u a nd we rep eat the a rgumen t with G − u if it is twin-free. In this case if G − u ∈ { K 1 ,t | t ≥ 2 } ∪ A , we apply Case 1 or Case 2. If G − u ∈ A ⊲ ⊳ K 1 with u ′ being the vertex of K 1 , then u a nd u ′ induce an iso morphic copy of A 1 and G ∈ A . If G − u is no t t win-free then, by L e mma 5, x must b e one o f the twin vertices. Let x ′ be its t w in and suppo se x ′ ∈ V ( A i 1 ) with V ( A i 1 ) = { z 1 , z 2 , . . . , z 2 k } . Without lo ss of gener alit y we may assume x ′ = z l with l ≤ k . If l ≥ 2 , then we claim C = V ( G ) \ { z l , z 2 k } is an identifying co de of G which is a contradiction. T o prov e our c la im notice first that vertices of A i 2 ⊲ ⊳ · · · ⊲ ⊳ A i j are a lr eady identified from each other and from the other vertices. Now each pair of vertices of A i 1 is sepa r ated b y a vertex in V ( A i 1 ) ∩ C except z l + k − 1 and z l + k which ar e separated by x . The vertex x is also separated from all the other v ertices by u . It remains to show that u is separa ted from vertices of A i 1 . It is separated from vertices in { z 1 , . . . , z l + k − 1 } by x a nd from { z k +1 , . . . , z 2 k } by z 1 ( l ≥ 2) . Thus x ′ = x 1 and now it is easy to see that the subgra ph induced by V ( A i 1 ) , u a nd x is isomorphic to A i 1 +1 and, therefo re, G ∼ = A i 1 +1 ⊲ ⊳ A i 2 ⊲ ⊳ . . . ⊲ ⊳ A i j . Since every graph in { K 1 ,t | t ≥ 2 } ∪ A ∪ ( A ⊲ ⊳ K 1 ) has ma xim um degree n − 2 , we hav e: Corollary 15. L et G b e a twin-fr e e c onne cte d gr aph on n ≥ 3 vertic es and maximum de gr e e ∆ ≤ n − 3 . Then γ ID ( G ) ≤ n − 2 . 4. Infinite graphs It is shown in [8] that Theorem 1 do es not hav e a dire c t e x tension to the fa mily of infinite graphs. In other words, there a re nontrivial examples of twin-free infinite graphs requiring the whole vertex set fo r any iden tifying co de. The basic example of s uc h infinite gr aphs, origina lly defined in [8], is g iv en b elow. In this section, we cla ssify all such infinite g raphs. This strengthens a theorem of [12], which claims that there are no s uc h infinite graphs in whic h all vertices hav e finite deg rees. Definition 16. L et X = { . . . , x − 1 , x 0 , x 1 , . . . } and Y = { . . . , y − 1 , y 0 , y 1 , . . . } . A ∞ = ( X ∪ Y , E ) is t he gr aph on X ∪ Y having e dge set E = { x i x j | i 6 = j } ∪ { y i y j | i 6 = j } ∪ { x i y j | i < j } . 6 ... x − 2 x − 1 x 0 x 1 x 2 ... ... y − 2 y − 1 y 0 y 1 y 2 ... Infinite clique o n X Infinite clique o n Y Figure 2: The graph A ∞ whic h needs all its ver tices f or an y iden tifying co de See Figure 2 fo r a n illustration. It is shown in [8] tha t the only separating set of A ∞ is V ( A ∞ ) . O ne should no te that the g raph induced b y { y 1 , y 2 , . . . , y k , x 1 , x 2 . . . , x k } is isomorphic to the graph A k . Before int ro ducing our theorem let us see ag ain wh y every sep a rating set of A ∞ needs the whole vertex set: for every i , x i and x i +1 are only separated by y i +1 , while y i and y i +1 are sepa rated only b y x i . This prop erty would still hold if we a dd a new vertex whic h is adjacent either to all vertices in X (similarly in Y ) or to no ne. T his leads to the following family: Let H be a finite or infinite simple g raph with a perfect matching ρ , that is a mapping x → ρ ( x ) o f V ( H ) to itself s uch that ρ 2 ( x ) = x and xρ ( x ) is an edge of H . W e define Ψ ( H, ρ ) to b e the graph built as follows: for every vertex x of H we assign Φ( x ) = { . . . x − 1 , x 0 , x 1 , . . . } . The vertex s et of Ψ( H , ρ ) is [ x ∈ V ( H ) Φ( x ) . F or each edge xρ ( x ) of H we build a copy o f A ∞ on Φ( x ) ∪ Φ( ρ ( x )) and for every other edge xy o f H we join every v ertex in Φ( x ) to every vertex in Φ( y ) . An example of such construction is illustrated in Figur e 3. x 1 y 1 y 2 x 2 y 3 x 3 H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 } Ψ − → A ∞ Y 1 X 1 A ∞ Y 3 X 3 ⊲ ⊳ ⊲ ⊳ A ∞ X 2 Y 2 Figure 3: Construction of Ψ( H, ρ ) from ( H, ρ ) W e now hav e: Prop osition 17. F or every simple, finite or infinite, gr aph H with a p erfe ct matching ρ , the gr aph Ψ( H , ρ ) c an only b e identifie d with V (Ψ( H , ρ )) . 7 Pr o of. Let A x be the copy of A ∞ which c o rresp onds to the edge xρ ( x ) . Then for every vertex y in V (Ψ( H , ρ )) \ V ( A x ) , either y is c onnected to every vertex in A x or to neither of them. Thus to separ ate vertices in A x , we need a ll the v ertices o f A x . Since x is arbitra ry , we need all the vertices in V (Ψ( H, ρ )) in any se pa rating set. In the next theorem we prov e that every such extremal connected infinite gra ph is Ψ( H , ρ ) for some connected finite or infinite gra ph H together with a matching ρ . Theorem 18. L et G b e an infinite c onne cte d gr aph. Then a pr op er subset C of V ( G ) identifies al l p airs of vertic es of G unless G = Ψ( H , ρ ) for some fin ite or infin ite gr aph H to gether with a p erfe ct matching ρ . Pr o of. W e already hav e seen that if G ∼ = Ψ( H , ρ ) , then the only identifying co de of G is V ( G ) . T o prove the conv er s e supp o se G − v ha s a pair of twin vertices for every vertex v of G . It is enough to s ho w that every vertex v of G belong s to a unique induced subgr aph A v of G isomorphic to A ∞ and that if a vertex not in A v is adjacent to a vertex in the X (respectively , Y ) part o f A v then it is a djac e nt to a ll the vertices of the X (r e s pectively , Y ). Let x 1 be a vertex of G . The subgraph G − x 1 has a pair of t wins, let y 1 and y 2 be o ne such pair. Assume, without lo ss of generality , that x 1 is adjacent to y 2 and not to y 1 . By Lemma 5, x 1 m ust be o ne of the vertices o f a pair of twins in G − y 1 . Let the other be x 2 . Now consider the subgraph G − y 1 . This subgraph must have a pair of twins and x 1 m ust b e one of them. Let x 0 be the o ther one. Contin uing this pro cess in b oth directions (with negative and p ositive indices) we build our A x 1 ∼ = A ∞ as a subg r aph of G . Since each consecutive pair of vertices in X ⊂ A x 1 is separated o nly by a vertex in Y ⊂ A x 1 , every pair of vertices in X are twins in G − Y . Th us each vertex no t in A x 1 , either is adjacent to all the v ertices in X or to none o f them. Similarly , every vertex in A x 1 , either is adjac e n t to all the vertices in Y or to none. Hence A x 1 is unique. This prov es the theorem. 5. Bounding γ ID ( G ) by n and ∆ In this section, we intro duce new upper b ounds on pa rameter γ ID in ter ms of b oth the order and the maximum degree of gr aph, th us extending a result o f [12]. W e define A + ∞ to b e the subgr aph of A ∞ induced by the vertices of p o sitiv e indices in X and in Y . The following lemma, which is a strengthening of Theorem 1, has b een attributed to N. Bertr and [2]. W e give an indep endent pro of a s [2] is not accessible. Lemma 19 ([2]) . If G is a t win-fr e e gr aph (infinite or not) not c ontaining A + ∞ as an induc e d s u b gr aph, then for every vertex x of G , ther e is a vertex y ∈ B 1 ( x ) such that G − y is twin-fr e e. Pr o of. By co n tra diction, suppose that x 1 is a vertex that fails the statement of the lemma. Then G − x 1 has a pa ir of twin vertices. W e name them y 1 and y 2 . Without loss of generality we assume that x 1 is adjacent to y 2 but not to y 1 . Now, in G − y 2 we must ha ve another pair u, u ′ of twin v ertices. By Lemma 5, x 1 ∈ { u, u ′ } , we name the other element x 2 ( x 2 ∈ B 1 ( x 1 ) ). Note that the subgr aph induced on x 1 , x 2 , y 1 , y 2 is isomo rphic to A 2 . W e prov e by induction that A + ∞ is a n induced subgraph of G , th us obtaining a co n tr adiction. T o this end supp ose A k on { y 1 , . . . y k , x 1 , . . . , x k } is a lready built such that x k − 1 , x k are twins in G − y k and y k − 1 , y k are twin s in G − x k − 1 . T hen x k ∈ B 1 ( x 1 ) . Co nsider G − x k . T her e m ust b e a pair of twins and, b y Lemma 5, y k m ust be one of them. Let y k +1 be the other o ne. Since y k and y k +1 are t wins in G − x k , then y k +1 is adjacent to x 1 , . . . , x k and y 1 , . . . , y k , in particular y k +1 ∈ B 1 ( x 1 ) . Now, ther e mu st b e a pair of twins in G − y k +1 and again by Lemma 5 one of them must b e x k , let the other o ne be x k +1 . Since x k and x k +1 are twins in G − y k +1 , then x k +1 is adjacent to x 1 , . . . , x k and not adjacent to y 1 , . . . , y k . Thus the graph induced on { y 1 , . . . , y k +1 , x 1 , . . . , x k +1 } is isomor phic to A k +1 with the pro perty that x k , x k +1 are t wins in G − y k +1 and y k , y k +1 are twins in G − x k . Since this pro cess do es not end, we find that A + ∞ is a n induced subgraph o f G . It w as conjectured in [1 0] that: 8 Conjecture 20 ([10]) . F or every c onne cte d twin-fr e e gr aph G of maximum de gr e e ∆ ≥ 3 , we have γ ID ( G ) ≤ l | V ( G ) | − | V ( G ) | ∆( G ) m . In s upport of this conjecture, we prov e the following weaker upper b o und on the size of a minim um iden tifying co de of a t win-free g raph. W e note tha t a similar b ound is pr o ved in [10]. Theorem 21. L et G b e a c onne cte d, twin-fr e e gr aph on n vertic es and of maximum de gr e e ∆ . Then γ ID ( G ) ≤ n (1 − ∆ − 2 ∆(∆ − 1) 5 − 2 ) = n − n Θ(∆ 5 ) . Pr o of. First, we note that if I is a maximal 6-indep endent set, then | I | ≥ n (∆ − 2) ∆(∆ − 1) 5 − 2 . This is true b ecause | B 5 ( x ) | ≤ ∆(∆ − 1) 5 − 2 ∆ − 2 for every vertex x . No w, let I b e a 6-indep endent set. F or each vertex x ∈ I let f ( x ) be the vertex found using Lemma 19 and f ( I ) = { f ( x ) | x ∈ I } . Since I is a 6 -independent set, f ( I ) is a 4-indep enden t set of G and | f ( I ) | = | I | . Now, by Lemma 7, we know that C = V ( G ) \ f ( I ) is an iden tifying co de of G . The b ound is now obtained by taking any maxima l 6-indep endent s e t I . It is ea sy to observe that if G is a regular t win-free gr aph, then V ( G ) − x is a n iden tifying co de for every vertex x of G . Th us the res ult of theorem 21 can b e slightly improv ed for regular graphs as follows: Theorem 22. L et G b e a c onne cte d ∆ -re gular twin-fr e e gr aph on n vertic es. Then γ ID ( G ) ≤ n (1 − 1 1+∆ − ∆ 2 +∆ 3 ) = n − n Θ(∆ 3 ) . Pr o of. W e note that a 4 -independent set I of size a t least n 1+∆ − ∆ 2 +∆ 3 can b e found b ecause | B 3 ( x ) | ≤ ∆(∆ − 1) 3 − 2 ∆ − 2 = 1 + ∆ − ∆ 2 + ∆ 3 . Now, G − x is t win-free for every vertex x of I (b ecause G is regular ), so b y L emma 7, V ( G ) − I is an identif ying co de of G . It is prov ed in [12] that in any nont rivial infinite t win-free g raph G whose vertices are all of finite degree, there exists a vertex x such that V ( G ) \ { x } is a n identifying co de of G . Using Lemma 19 and similar to the pro of of Theorem 21, we can s trengthen their result as follows: Theorem 23. L et G b e a c onne cte d infi nite twin-fr e e gr aph whose vert ic es al l have fin ite de gr e e. Then ther e exists an infinite set of vertic es I ⊆ V ( G ) , such that V ( G ) \ I is an identifying c o de of G . 6. General r -identifying co des T o iden tify the clas s of graphs w ith γ ID r ( G ) = n − 1 one needs to find the r -ro ots of the g raphs in { K 1 ,t | t ≥ 2 }} ∪ A ∪ ( A ⊲ ⊳ K 1 ) . The general problem of finding the r -ro ot o f a gr aph H is an NP-hard problem [16] and it do es not seem to b e an easy tas k in this particular case either. If s divides k − 1 and r = k − 1 s , then the g raph G = P s 2 k is one of the r -ro ots of A k . It is e a sy to see that, in most c a ses, one can remove many edg es of G and still have G r ∼ = A k . T he difficulty of the pro blem is that an r -r oot o f A k is not necess arily a subgraph of P s 2 k . An example of s uc h a 2-ro ot of A 5 is given in Figure 4. 1 2 3 4 5 6 7 8 9 10 Figure 4: A 2-ro ot of A 5 whic h is not a subgraph of P 2 10 F or the case of infinite gra phs , we note that there exists a 2-ro ot of A ∞ . T his gra ph is defined a s follows: it has the same v ertex set X ∪ Y as A ∞ and the same edges be t ween X and Y , but no edges within X or Y . Ho wev er, we do not know whether there exist other ro ots of g r aphs describ ed in Theor em 18. W e should also note that a (3 r + 1) -indep enden t set in G r is a 4-indep endent set in G . Thus we ha ve the following g eneral form of Lemma 7, Theo rem 21 and Theorem 23: 9 Lemma 24. L et G b e a c onne cte d gr aph on n vertic es such that G r is twin-fr e e. L et I b e a (3 r + 1) - indep en dent set of G such that for every vertex v of I the set V ( G ) \ { v } is an r -identifying c o de of G . Then C = V ( G ) \ I is an r -identifying c o de of G . Theorem 25. Le t G b e a c onne ct e d gr aph on n vertic es and of maximum de gr e e ∆ su ch that G r is t win-fr e e. Then γ ID r ( G ) ≤ n (1 − ∆ − 2 ∆(∆ − 1) 5 r − 2 ) = n − n Θ(∆ 5 r ) . Theorem 26 . L et G b e a c onne cte d infinite gr aph whose vert ic es ar e of fin ite de gr e e such t hat G r is t win- fr e e. The n ther e exists an infinite set of vertic es I ⊆ V ( G ) , such t hat V ( G ) \ I is an r -identifying c o de of G . 7. Remarks W e conclude o ur pap er b y s o me remar k s on related works. Remark 1 The following t wo questions were p osed in [18]: 1. Do there e x ist k -reg ular g raphs G of order n w ith γ ID ( G ) = n − 1 for k < n − 2 ? 2. Do there e x ist gr aphs G of o dd order n and maximum deg ree ∆ < n − 1 with γ ID ( G ) = n − 1 ? As a coro llary o f Theorem 14, we can now a nsw er these questions in the negative. Indeed, for the firs t question, if G is a k -regular ( k ≥ 2 ) g raph of order n with γ ID ( G ) = n − 1 then G is the join of k disjoin t copies of A 1 . F or the second question, noting that ea c h g raph in A has a n even o rder, we co nclude that if a graph G on an o dd num ber , n , of vertices has γ ID ( G ) = n − 1 , then G ∈ { K 1 ,t | t ≥ 2 } ∪ ( A ⊲ ⊳ K 1 ) and, therefore ∆( G ) = n − 1 . Remark 2 Given a graph G = ( V , E ) the 1-b al l memb ership gr aph of G is defined to be the bipartite graph G ∗ = ( I ∪ A, E ∗ ) where I = V ( G ) , A = { B 1 ( x ) | x ∈ V ( G ) } and E ∗ = {{ u , B 1 ( v ) } | u ∈ B 1 ( v ) , u, v ∈ V ( G ) } . It is no t hard to s ee that the problem of finding iden tifying c odes in G is equiv alent to the o ne of finding dis c riminating co des in G ∗ . But since no t every bipartite graph is a 1-ball mem b ership gra ph, the latter co n tains the former pro perly . It is a rephrasing of Bondy’s theor em [1], that every bipartite g r aph ( I ∪ A, E ) has a discr imina ting co de of siz e at mo s t | I | . 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