Bounding the Feedback Vertex Number of Digraphs in Terms of Vertex Degrees

Bounding the Feedback V erte x Number of Digraphs in T erms of V erte x De grees Hermann Gruber knowle dgepa rk AG Leonr odst r . 68, M ¨ unch en, Germany Abstract The T ur ´ an bound [17] i s a famous result in graph theory , which relates the independen ce number of an undirected graph to its edge density . Also the Caro–W ei inequ ality [4, 18], which gi ves a more refined bound in terms of the vertex degree sequ ence of a grap h, might be r e garded today as a classical result. W e show h o w these statements can be generalized to dire cted graph s , thu s yielding a boun d on directed feedback vertex number in terms of vertex outdegrees and in terms of av erage outdegree, respecti vely . K e ywor d s: directed feedback vertex numb er , Caro–W ei inequality , feedb ack set, acyclic s et, induced D A G 2000 MSC: 05C20 , 68R10 1. Introduction Not only in discrete mathematics, generalizing existing concepts and proof s has always been a gu iding theme for researc h. T he great mathematician Henri Poin car ´ e even con sidered this a s the leitmotiv of all mathem atics. 1 In particular , many results from graph theory were gen eralized to weighted graphs, digraphs, or h ypergraphs. So metimes, providing such ge neralizations is an easy exercise; in oth er cases, the main di ffi culty lies in form ulating the “rig ht generalization” of the orig inal the orem. An additional obstacle is imp osed if the result we intend to generalize allows for several proofs or equiv alent r eformulations. Then there are many roads to potential generalization s to explore, a nd selecting the most pro mising o ne can be di ffi cu lt. Howe ver , once the pro per gen eralizations of the used notions are foun d, the more general proof often runs very much along the same l ines. As we shall see, one such example is the T ur ´ an bo und [17], which gives the number of edg es that a graph of order n can have when fo rbidding k -c liques as s ubgrap hs. It allo ws for many dif- ferent proo fs and equivalent r eformulations, see [1]. A d ual v ersion of T ur ´ an’ s bou nd, regarding the size of independe nt sets, was refined by Caro [4] and W ei [18]. Their result has subsequently been gen eralized, by replacing the indep endent sets with less restricte d in duced subgrap hs [3], Email addr ess: info@he rmann-gr uber.com (Hermann Gruber) 1 Poincar ´ e’ s original phrasing was more poetic: “La math ´ ematique est l’ar t de donner le m ˆ eme nom ` a des choses di ff ´ erentes. ” [14, p. 29]. Prep rint submitted to Discr ete Applied Mathema tics Nove mber 20, 2018 respectively by replacin g the conc ept of a graph with m ore general n otions, nam ely weighted graphs [16] and hypergrap hs [5]. Here we comple ment these e ff orts by providing a generaliza- tion of th e Caro–W ei bo und to the case of dig raphs. From an algorithm ic perspective, the new result gaug es a simp le g reedy heu ristic fo r the min imum d irected f eedback vertex set pro blem. In this way , the main result of th is paper yields a formalized cou nterpart to the intuition that the minimum (directed ) feedback vertex num ber of sparse digraphs cannot be “overly large”. 2. Preliminaries W e assum e the reader is familiar with ba s ic no tions in the theory of digr aphs, as con tained in textbo oks such as [10]. Nev ertheless, we briefly reca ll the most impo rtant notio ns in the following. A digraph D = ( V , A ) consists of a finite set, referred to as th e set V ( G ) = V of vertices , and of an irre fle xi ve binary relation o n V ( G ) , referred to as the set o f ar cs A ( G ) = A ⊂ V × V . Th e cardinality of the vertex set is refer red to as the or d er of D . In the s pecial case where the arc relation of a digraph is symmetric, we also speak of an (undir ected) graph . For a v ertex v in a digr aph D , define its out-n eighborhood as N + ( v ) = { u ∈ V | ( u , v ) ∈ A , u , v } , an d its o ut- de gr ee as d + ( v ) = | N + ( v ) | . I n-neighbor hood an d in-degree are d efined analogou sl y , and den oted by N − ( v ) and d − ( v ), respectively . The degr ee d ( v ) of v is then de fined as d ( v ) = | N − ( v ) ∪ N + ( v ) | , and the total de gr ee of v is d efined as | N − ( v ) | + | N + ( v ) | . W e note that ou r definition of vertex degree agrees (on u ndirected graph s) with the standard usage of this no tion in the theor y of undirected graphs, see e. g. [7]. For a subset o f vertices U ⊆ V of the digr aph D = ( V , A ), the subd igr aph induced by U is the dig raph ( U , A | U × U ) obtained by reducing the vertex set to U an d by restricting the arc set to the relation induced b y A o n U . If a dig raph H c an be obtained in th is way by approp riate restriction of the vertex set o f th e dig raph D , we say H is an indu ced subd igr aph of D . A simple path in a digraph is a sequence of k ≥ 1 arcs ( v 1 , w 1 )( v 2 , w 2 ) · · · ( v k , w k ), such that for all 1 ≤ i < k hold s w i = v i + 1 and all start-vertices v i are d ist inct. I f fur thermore w k = v 1 , we speak of a cycle . In particular, notice th at each pair of op posite arcs ( v , w )( w , v ) in a digr aph amounts to a cycle. This convention is commonly used in the theory of digraph s, co mpare [10 ]. A digraph con taining no cycles is called acyc lic , o r a dir ected acyclic graph (D A G) . For a vertex sub set U of a digra ph D , if the subdigr aph induced by U is acyclic, then we call U an acyclic s et . In particular , if D [ U ] contains no arcs at all, then U is called an independent set . The maximum car dinality among all indepen dent sets in D is called the independ ence numb er of D . T u r ´ an proved the f ollo wing bou nd on the independ ence number of undirected graphs: Theorem 1. Let D = ( V , A ) be an undir ected graph of or d er n an d of aver age de gree d. Then D contains an indepen dent set of size at least  d + 1  − 1 · n. Caro [4], and, indepen dently , W ei [1 8 ] proved the following refined bound: Theorem 2. Let D = ( V , A ) be an undirected graph of or der n. Then D co ntains an ind ependent set of size at least P v ∈ V ( d ( v ) + 1 ) − 1 . A set F of vertices in a digraph D = ( V , A ) is called a feedback vertex set if V \ F is an acyclic set. The feedba c k vertex nu mber τ 0 ( D ) of D is defined a s the min imum cardin ality among a ll feedback vertex sets for D . A simple observation is that for a digraph D of order n , the cardin ality of a maximu m acyclic set equals n − τ 0 . 2 3. Directed F eedback V ertex Sets and V ert ex Degrees Quite recen tly , se veral new a lgorithms wer e devised fo r exactly solving the min imum di- rected f eedback vertex set pr oblem [6, 15]. But all k no wn exact algo rithms fo r this pro blem share the undesirable feature that their worst-case runn ing time is expon ential—in th e ord er n of the input graph , o r at least in the size of the feedbac k vertex numb er τ 0 . Th is is not surprising as the pro blem has been known for a long time to be NP -co mplete, see [8]. Here, we consider the f ollo wing simple greedy heuristic fo r fin ding a large acyclic set, and hence a small feedback vertex set, in a digraph D . W e call th e algorith m M in-Gr eedy , in acc or - dance with a homonym ous greedy heuristic on undirected grap hs f or finding a l arge indep endent set, compar e [9, 11]. Starting with D 1 = D , we in ducti vely define a sequenc e of digraphs D i , i ≥ 1, b y first choosing a vertex v i , such tha t v i has minimu m outdegree in D i , and then deleting v i , alon g with its out-n eighborhoo d in D i , to obtain the digrap h D i + 1 . W e pro ceed in doing so until the vertex set of D i is empty , and reme mber the vertices v i selected in each turn . These vertices fo rm the set S = { v 1 , v 2 , . . . v r } , which is the result finally returned by the procedu re. Before we analyze the quality of the above heu ristic, we shall first prove its soun dness. Lemma 3. Let D be a digr aph. Then the set S returned by Min-Greed y on in put D is an acyclic set in D. Pr oo f . Using the notions fro m the de scription o f th e alg orithm, it su ffi ces to show that fo r all v j , v k ∈ S , th e condition j < k implies that the digraph D has no arc ( v j , v k ). This claim implies that a long ev ery simp le path in D [ S ] = D [ { v 1 , v 2 , . . . , v r } ], th e vertex ind ices must ap pear in decreasing order, thus ruling out the possibility of a cycle in D [ S ]. T o prove the claim, observe first that for all k > j , D k is an induced subdig raph of D j + 1 . Th us starting with D j + 1 = D j − ( { v j } ∪ N + ( v j )), no vertex in the out-neighb orhood of v j is present in any of the subsequent digraphs. But v k is selected from G k , hence is present in G k and cannot be in the out-n eighborhood of v j . Observe that the pro of of Lemma 3 do es n ot d epend on the choice of a vertex of minimum degree in D i for v i —the algorith m is sound if we c hoose any vertex in D i for v i . Now we are ready to state our main result. Theorem 4. Let D = ( V , A ) be a digraph of or der n. Then Min-Gr eedy always fin ds an acyclic set in D of size at least P v ∈ V ( d + ( v ) + 1 ) − 1 . Pr oo f . Using the notation fro m the algo rithm, let v i be the s elected vertex of minimum o utdegree in D i . Th en for all vertices w ∈ N + D i ( v i ) ∪ { v i } holds d + D ( w ) + 1 ≥ d + D i ( w ) + 1 ≥ d + D i ( v i ) + 1 =    N + D i ( v i ) ∪ { v i }    . Thus, X w ∈ N + D i ( v i ) ∪{ v i }  d + D ( w ) + 1  − 1 ≤ X w ∈ N + D i ( v i ) ∪{ v i }  d + D i ( w ) + 1  − 1 ≤ X w ∈ N + D i ( v i ) ∪{ v i }     N + D i ( v i ) ∪ { v i }     − 1 = 1 . 3 On the other h and, since the algorithm partition s the vertex set of D into a disjo int union of subsets as V ( D ) = | S | [ i = 1  N + D i ( v i ) ∪ { v i }  , we have X v ∈ V  d + D ( v ) + 1  − 1 = | S | X i = 1 X w ∈ N + D i ( v i ) ∪{ v i }  d + D ( w ) + 1  − 1 ≤ | S | X i = 1 1 = | S | , as desired. Just like the Caro–W ei bo und [4, 18] for the independ ence numbe r of undirec ted graph s implies the T u r ´ an bou nd [17] by th e inequality of arithmetic an d harmon ic means, we have the fo llo wing simp le b ound on the size o f a m aximum acyclic set, and hence, on the directed feedback vertex numb er , in terms of a verage outdegree: Corollary 5. Let D = ( V , A ) be a digr aph of or d er n and of average outd e gree d + . Then τ 0 ( D ) ≤ n ·  1 −  d + + 1  − 1  . Pr oo f . W e show the equivalent statement that the digrap h D has an acyclic set of cardinality at least n /  d + + 1  . The boun d P v ∈ V ( d + ( v ) + 1 ) − 1 from Theorem 4 looks di ff erent a t first glance. Nev ertheless, it easily implies a bound in terms of average o utdegree: recall that the ineq uality of the ha rmonic, geometric and arithmetic mean (see [2, Chapter 16]) states that the geo metric mean of n positive number s a 1 , a 2 , . . . , a n is sandwic hed between the harmon ic m ean and the arithmetic mean of these num bers, that is, n P n i = 1 a − 1 i ≤        n Y i = 1 a i        1 / n ≤ n X i = 1 a i / n . Now cho ose the a i to b e the vertex degrees in D incre ased by 1 each. Then the outer most inequality y ields n P v ∈ V ( d + ( v ) + 1 ) − 1 ≤ P v ∈ V ( d + ( v ) + 1) / n . A very simple calcu lation co mpletes the proof . Both th e bound fro m T heorem 4 and the one from Corollary 5 are sharp, as witnessed, fo r example, by the digrap h of order k · m that is obtained as the disjoint union of m many k -cliqu es. Notice that we obtain the Caro–W ei bou nd and the T ur ´ an bound, respectively , if we restrict the sco pe of th e above statements to symmetric dig raphs: fo r these, th e size of a maximum acyclic set is equa l to the in dependence number, and the o utdegree of each vertex is equal to its degree (which in turn is equal to half its total degree). As a final note, we r emark that the Moon–Moser bound on the numb er of maxim al indep en- dent sets i n undirected grap hs [13] does not generalize to an analogous statement about maximal acyclic sets; as a matter of f act, not e ven tournam ents allow f or a clear generalization [12]. In the undirected case, the proofs of both the Caro–W ei bo und and the Mo on–Moser bou nd can be u sed to derive Tur ´ an’ s graph theorem, see [1]. A general theme for further r esearch is to identify those fragmen ts of the theory of undirected graphs that generalize smoothly to the case of digraphs. 4 References [1] M. Aigner , Tur ´ an’ s Graph Theorem, The American Mathematic al Monthl y 102 (9) (1995) 808–816. [2] M. Aigner , G. M. Ziegler , Proofs from THE BOOK, Springer , 2nd edn., 2001. [3] N. Alon, J. Kahn, P . D. S e ymour , L a rge Induced Degenera te Subgraphs, Graphs and Combinatoric s 3 (1) (1987) 203–211. [4] Y . Caro, New Results on the Independe nce Number , T ech. Rep., T el A vi v Uni ve rsity , 1979. [5] Y . Caro, Zs . T uza, Hypergraph Coveri ngs and L oc al Colorings, Journal of Combinatoria l Theory , Series B 52 (1) (1991) 79–85. [6] J . Chen, Y . Liu, S. L u, B. O’Sulli van, I. Razgon, A Fix ed-Pa rameter Algorithm for the Direc ted Feedbac k V ertex Set Problem, Journal of the A CM 55 (5) (2008) Article No. 21. [7] R. Diestel, Graph Theory , vol. 173 of Graduate T exts in Mathe matics , Springer , 3rd edn., 2006. [8] M. R. Gare y , D. S. Johnson, Computers and Intractabili ty: A Guide to the Theory of NP-Completene ss, A Serie s of Books in the Mathemati cal Scie nces, W . H. Freeman, 1979. [9] J . R. Griggs, Lower Bound s on the Independen ce Number in T erms of the Degrees, Journal of Combinatorial Theory , Series B 34 (1) (1983) 22–39. [10] G. Gutin, J. Bang-Jense n., Digraphs: Theory , Algorithms and Applica tions, Springer , 1st edn., 2000. [11] M. M. Halld ´ orsson, J. Radhak rishnan, Greed is Good: Approximatin g Inde pendent Sets in Sparse and Bounded - Degre e Graphs., Algorithmica 18 (1) (1997) 145–163. [12] J. W . Moon, On Maximal Transiti ve Subtou rnaments, Proceedings of the Edinb urgh Mathe matical Society (Seri es 2) 17 (4) (1971) 345–349. [13] J. W . Moon, L. Moser , On Cliques in Graphs, Israel Journal of Mathematics 3 (1) (1965) 23–28. [14] H. Poincar ´ e, Science et M ´ ethode, Flammarion, 1908. [15] I. Razgon, Computing Minimum Directed Feedbac k V ertex Set in O ∗ (1 . 9977 n ), in: G. F . Italia no, E. Moggi, L. Laura (Eds. ), Proceedi ngs of the 10th Ital ian Confe rence on Theoret ical Comput er Science, W orld Scie ntific, 70–81, 2007. [16] S. Sakai, M. T ogasak i, K. Y amazaki, A Note on Greedy Algorit hms for the Maximum W eigh ted Independent Set Problem, Discrete Applied Mathemat ics 126 (2-3) (2003) 313–322. [17] P . T ur ´ an, On an E xt remal Problem in Graph Theory (in Hungarian), Matematikai ´ es Fizikai Lapok 48 (1941) 436–452. [18] V . K. W . W ei, A Lowe r Bound on the Stabil ity Number of a Simple Gra ph, T echnic al Memorandum No. 81-1121 7- 9, Bell Laboratori es, 1981. 5