BI-INDUCED SUBGRAPHS AND STABILITY NUMBER *

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1 Yugoslav Joural of Operatios Research 14 (2004), Number 1, BI-INDUCED SUBGRAPHS AND STABILITY NUMBER * I E ZVEROVICH, O I ZVEROVICH RUTCOR Rutgers Ceter for Operatios Research, Rutgers Uiversity, Piscataway, New Jersey, USA igor@rutcorrutgersedu Received: Jue 2003 / Accepted: February 2004 Abstract: We defie a 2-parametric hierarchy CLAP ( m, ) of bi-hereditary classes of graphs, ad show that a maximum stable set ca be foud i polyomial time withi each class CLAP ( m, ) The classes ca be recogized i polyomial time Keywords: Stability umber, hereditary class, bi-hereditary class, forbidde iduced subgraphs, forbidde bi-iduced subgraphs 1 INTRODUCTION A set S V( G) i a graph G is stable (or idepedet) if S does ot cotai adjacet vertices A stable set of a graph G is called maximal if it is ot cotaied i aother stable set of G A stable set of a graph G is called maximum if G does ot have a stable set cotaiig more vertices The cardiality of a maximum stable set i G is the stability umber of G, ad it is deoted by α ( G) Decisio Problem 1 (Stable Set) Istace: A graph G ad a iteger k Questio: Is there a stable set i G with at least k vertices? This problem is kow to be NP-complete (Karp [7], see also Garey ad Johso [3]) A class P of graphs is α-polyomial if there exists a polyomial-time algorithm to solve Stable Set Problem withi P We shall defie a hierarchy CLAP ( m, ) of α-polyomial graph classes The hierarchy covers all graphs * The first author was supported by DIMACS Witer 2002/2003 Award AMS Subject Classificatio: 05C69

2 28 IE Zverovich, OI Zverovich / Bi-Iduced Subgraphs ad Stability Number Note that it is easy to fid the stability umber of graphs i ay class without large coected iduced bipartite subgraphs I other words, the class CONNBIP( N) - free graphs is α-polyomial, where CONNBIP ( N) is the set of all coected bipartite graphs of order N Lozi ad Rautebach [8] used this fact to produce α-polyomial subclasses of CONNBIP( N) -free graphs defied by a path ad a star as forbidde subgraphs Specifically, give m ad, there exists a iteger N such that each ( P, K1, m) -free triagle-free graph is a CONNBIP( N) -free graph I our hierarchy we also forbid a path, but we do ot forbid a star Istead, we use Hall's theorem to specify a particular family of coected bipartite graphs, thus obtaiig a more geeral result 2 BI-INDUCED SUBGRAPHS The eighborhood of a vertex x i a graph G is deoted by N( x) = NG ( x) For a subset X of VG, ( ) we deote N( X) = N ( x) x X G Defiitio 1 A bipartite graph F is called a bi-iduced subgraph of a graph G if (BI1): F is a subgraph of G [ot ecessarily iduced], ad (BI2): there exists a bipartitio A B of V(F) such that both A ad B are stable sets i G I other words, a bi-iduced subgraph F of a graph G is obtaied from a bipartite iduced sub graph F' of G by deletig some edges [possibly, oe] As usual, we distiguish bi-iduced subgraphs up to isomorphism A class P is bi-hereditary if it is closed uder takig bi-iduced subgraphs That is, F P wheever G P ad F is a bi-iduced subgraph of G Clearly, a class is bihereditary if ad oly if it ca be characterized i terms of forbidde bi-iduced subgraphs Also, a bi-hereditary class with fiitely may miimal forbidde bi-iduced subgraphs ca be recogized i polyomial time We defie a 2-parametric series CLAP ( m, ) of bi-hereditary classes of graphs As usual, P deotes the -vertex path A m-claw is a complete bipartite graph of the form K If we subdivide every edge of a m-claw by a vertex, we obtai a bipartite graph of order 2m + 1 called a subdivided m-claw, SK (see Figure 1) Figure 1: Subdivided m-claw SK

3 IE Zverovich, OI Zverovich / Bi-Iduced Subgraphs ad Stability Number 29 Defiitio 2 Give itegers m 1 ad 1, the class CLAP ( m, ) cosists of all graphs that do ot cotai SK as bi-iduced subgraphs, ad P as iduced subgraphs Clearly, CLAP ( m, ) CLAP ( m+ 1, ), CLAP ( m, ) CLAP ( m, + 1) for all m 1 ad 1, ad CLAP ( m, ) m= 1= 1 cotais all graphs Note that membership i each CLAP ( m, ) ca be checked i polyomial time, sice there is oe miimal forbidde iduced subgraph ad there is oe miimal forbidde bi-iduced subgraph for this class 3 STABILITY IN CLAP ( m, ) Here is our mai result Theorem 1 For all itegers m 1 ad 1, the class CLAP ( m, ) is α -polyomial Proof: We defie N = N( m, ) = ( m+ 2) ( m+ 1) 2 d 1 (1) d = 1 if 3, ad N = 1 if 2 Now we apply the followig algorithm to a arbitrary graph G CLAP ( m, ) Algorithm 1 Step 0 Set S = 0 Step 1 For every stable set T V( G)\ S with T N, defie S' = ( S \ N( T)) T If S' > S, set S = S' Step 2 Retur S ad Stop The algorithm rus i polyomial time, sice N is a costat It produces a set S V( G) Suppose that S is ot a maximum stable set Claim 1 S is a stable set i G, ad there exists a stable set T V( G)\ S with T > N

4 30 IE Zverovich, OI Zverovich / Bi-Iduced Subgraphs ad Stability Number Proof: Iitially, S = 0 is a stable set Also, the set S' = ( S \ N( T)) T [o Step 2] is stable Thus S is a stable set i G Sice S is ot a maximum stable set, there exists a stable set I i G with I > S We deote T = I \ S Sice S < I we have S \ I < T, ad therefore NT ( ) S S\ I < T Step 1 of the algorithm implies that T Accordig to Claim 1, there exists a set T V( G)\ S such that (Tl): T > N, ad > N (T2): S' > S, where S' = ( S \ N( T)) T We assume that T has the miimum cardiality amog all sets that satisfy (Tl) ad (T2) Let H be a bipartite graph iduced by T U, where U = S \ I Claim 2 (i) For every vertex u T, there exists a matchig M i H - u that covers U, ad (ii) T = U + 1 Proof: (i) Each proper subset T' of T does ot satisfy (T2) [with T' istead of T] Ideed, if T' N, the it follows from Step 1 of the algorithm If T > N the it follows from miimality of T Let u T Each subset of T' = T \{ u} does ot have property (T2) I other words, for every X T ', we have N( X) X i H u By Hall's theorem (Hall [5], see also Hall [4]), there exists a matchig M i H u that covers T' I particular, T' U The coditio (T2) for T implies that T > U Therefore T' = U, ad M must cover U as well (ii) The statemet follows directly from (i) As usual, ( G) is the maximum vertex degree i G Claim 3 ( H) m+ 2 Proof: Suppose that there exists a vertex u V( H) of degree m + 2 First let u T Let u is adjacet to pairwise distict vertices v 1, v 2,, vm U By Claim 2(i), there exists a matchig M i H u that covers U We cosider the edges of M that are icidet to v1, v2,, v m Clearly, H u cotais SK as a hi-iduced subgraph Now let u U Let u is adjacet to pairwise distict vertices u1, u2,, um+ 2 T We apply Claim 2(i) to the graph H' = H u m + 2 : there exists a matchig M i H ' that covers U At most oe edge of M is icidet to the vertex u We see that H ' cotais SK as a hi-iduced subgraph It remais to ote that a hi-iduced subgraph i a iduced subgraph of G is also a hi-iduced subgraph of G

5 IE Zverovich, OI Zverovich / Bi-Iduced Subgraphs ad Stability Number 31 Note that Claim 2 implies coectedess of H Ideed, if H is ot coected the there is a compoet K i H such that oe part is larger tha the other, ad therefore deletig a vertex u T \ V( K) produces a graph without perfect matchig Claim 4 H cotais P as a iduced subgraph Proof: Accordig to (Tl), T N + 1 By Claim 2(ii), U = T 1 N Thus, V( H) 2N + 1 (2) If 2 the N = 1 ad 2N + 1= 3, ad the result follows Suppose that 3 Usig (2) ad (1), we obtai V( H) 2N ( m+ 2) ( m+ 1) The (3) ad Claim 3 imply V( H) 2 + ( 1) 2 d 1 (3) d = 1 2 d 1 (4) d = 1 Let u V( H) There are at most ( 1) d 1 vertices at distace d 1 from u Sice H is a coected graph, (4) implies that there exists a vertex v at distace 1 from u A shortest ( uv, )-path is a iduced P Claim 4 produces a cotradictio to the coditio that G CLAP ( m, ) This cotradictio shows that S is a maximum stable set i G Theorem 1 implies the followig results o α-polyomial classes: ( P5, K1, )- free graphs (Mosca [10]), a subclass of ( P5, K1,4) -free graphs (Brastädt ad Hammer [2]), ( P5, P, K2,3) -free graphs (Mahadev [9], see Figure 2), ad ( P 2 P 3, K 1, )-free graphs (Alekseev [1]) Figure 2: P 5, P ad K 2,3

6 32 IE Zverovich, OI Zverovich / Bi-Iduced Subgraphs ad Stability Number REFERENCES [1] Alekseev, VE, "O easy ad hard hereditary classes of graphs for the idepedet set problem", Discrete Appl Math (to appear) [2] Brastiidt, A, ad Hammer, PL, "O the stability umber of claw-free P 5 -free ad more geeral graphs", Discrete Appl Math, 95 (1-3) (1999) [3] Garey, MR, ad Johso, DS, Computers ad itractability A guide to the theory of NPcompleteess, W H Freema ad Co, Sa Fracisco, Calif, 1979 [4] Hall, M Jr, Combiatorial theory, Blaisdell Publ Co, Waltham, 1967 [5] Hall, P, "O represetatives of subsets", J Lodo Math Soc, 10 (1935) [6] Hertz, A, "Polyomially solvable cases for the maximum stable set problem", Discrete Appl Math, 60 (1-3) (1995) [7] Karp, RM, "Reducibility amog combiatorial problems", i: Complexity of computer computatios, Pleum Press, New York, 1972, [8] Lozi, V, ad Rautebach, D, "Some results o graphs without log iduced paths", RUTCOR Research Report RRR , RUTCOR, Rutgers Uiversity, 2003 [9] Mahadev, NVR, "Vertex deletio ad stability umber", Techical Report ORWP 90/2, Swiss Federal Istitute of Techology, 1990 [10] Mosca, R, "Polyomial algorithms for the maximum stable set problem o particular classes of P 5 -free graphs", Iform Process Lett, 61 (3) (1997)