A characterization of Konig-Egervary graphs using a common property of all maximum matchings

A c haracterization of K¨ onig–Egerv´ ary graphs using a common prop ert y of all maxim um matc hi ng s V adim E. Levit Ariel Univ ersit y Cen ter of Samaria, Ariel, Israel levitv@ariel.ac.il Eugen Mandrescu Holon Institute of T ec hnology , Holon, Israel eugen m@hit.ac.il Abstract The indep endenc e numb er of a graph G , denoted by α ( G ), is the cardinalit y of an i n dependent set of maximum size in G , while µ ( G ) is the size of a maximum matc h i ng in G , i. e., its matching numb er . G is a K¨ onig–Egerv´ ary gr aph if its order equals α ( G ) + µ ( G ). In this pap er w e give a new c haracterization of K ¨ onig–Egerv´ ary graphs. W e also deduce some prop erties of vertices b elonging to all maximum indep endent sets of a K¨ onig–Egerv´ ary graph. Key words: maximum indep endent set, maximum matching, core of a graph, critical vertex. 1 In tro d uction Throughout this pa p e r G = ( V , E ) is a simple (i.e., a finite, undirected, lo o pless a nd without multiple edges) graph with vertex set V = V ( G ) , edge set E = E ( G ), and order n ( G ) = | V ( G ) | . If X ⊂ V , then G [ X ] is the subgraph of G spanned by X . By G − W we mean the subgraph G [ V − W ], if W ⊂ V ( G ). F or F ⊂ E ( G ), by G − F we denote the pa rtial subgraph o f G obtained by deleting the edges of F , and w e use G − e , if W = { e } . If A, B ⊂ V and A ∩ B = ∅ , then ( A, B ) stands for the set { e = ab : a ∈ A, b ∈ B , e ∈ E } . The neighbo rho o d of a vertex v ∈ V is the s et N ( v ) = { w : w ∈ V , v w ∈ E } , and N ( A ) = ∪{ N ( v ) : v ∈ A } , while N [ A ] = A ∪ N ( A ) for A ⊂ V . By P n , C n , K n we mean the c hordles s path on n ≥ 3, the c ho r dless cycle o n n ≥ 4 vertices, a nd resp ectively the complete graph on n ≥ 1 vertices. 1 A set S of vertices is indep endent if no tw o vertices fro m S are adjacent. An in- depe ndent set of maximum size will b e referr ed to a s a maximum indep endent set o f G . The indep endenc e numb er of G , denoted b y α ( G ), is the cardinalit y of a maximum independent s et of G . By Ind( G ) we mean the s et of all indep endent sets of G . Let Ω( G ) denote the set of all max imu m indep endent se ts of G [15], and core( G ) = ∩{ S : S ∈ Ω( G ) } . A matching (i.e., a set of non-inc ide nt edges of G ) of max im um cardinality µ ( G ) is a maximum matching , and a p erfe ct matching is one cov ering all v ertices of G . A v e rtex v ∈ V ( G ) is µ -critic al provided µ ( G − v ) < µ ( G ). It is well-known that ⌊ n/ 2 ⌋ + 1 ≤ α ( G ) + µ ( G ) ≤ n hold for an y gra ph G w ith n vertices. If α ( G ) + µ ( G ) = n , then G is called a K¨ onig- Egerv´ ary gr aph (a K-E g raph, for shor t). W e attribute this definition to Deming [5], and Sterboul [25]. These graphs w ere studied in [3 , 11, 21, 22, 2 4], and generalized in [2, 2 3]. Several prop erties of K-E gra phs are presented in [14, 16, 17, 18, 19]. Theorem 1.1 [16] If G = ( V , E ) is a K¨ onig-Egerv´ ary gr aph, then: (i) e ach max imu m matching M of G m atches N ( c or e ( G )) int o c or e ( G ) ; (ii) H = G − N [ c or e ( G )] is a K -E gr aph with a p erfe ct matching and e ach maximum matching of H c an b e enlar ge d to a maximum matching o f G . According to a well-known result of K ¨ o nig [10] and Egerv´ ar y [7], every bipartite gr aph is a K-E graph. This cla ss includes also some non-bipa rtite graphs (see, for instance, the g raph fro m Figure 1). ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c u v x y G Figure 1: G is a K-E graph with α ( G ) = |{ a, b, c, x }| = 4 and µ ( G ) = |{ au, cv , xy }| = 3. It is easy to see that if G is a K- E graph, then α ( G ) ≥ µ ( G ) , and that a gr aph G having a pe r fect matc hing is a K-E gr aph if and o nly if α ( G ) = µ ( G ). If S is an indepe ndent set of a gra ph G and H = G [ V − S ], then we write G = S ∗ H . Clearly , any g r aph a dmits suc h representations. Ho wever, some particular cases are of sp ecial int erest. F or instanc e , if E ( H ) = ∅ , then G = S ∗ H is bipartite; if H is co mplete, then G = S ∗ H is a s plit gr aph [8]. Prop ositi on 1. 2 [16] If G is a gr aph, then the fol lowing assertions ar e e quivalent: (i) G is a K¨ onig-Egerv´ ary g r aph; (ii) G = S ∗ H , wher e S ∈ Ω( G ) and | S | ≥ µ ( G ) = | V ( H ) | ; (iii) G = S ∗ H , wher e S is an indep endent set with | S | ≥ | V ( H ) | and ( S, V ( H )) c ontains a matching M of si ze | V ( H ) | . 2 Let M b e a maximum matching of a gr aph G . T o adopt Edmonds’s ter minology , [6], we reca ll the follo wing terms for G relative to M . The edges in M are he avy , while those not in M are light . An alternating p ath fr om a vertex x to a vertex y is a x, y -pa th whose e dges a re alternating light and heavy . A vertex x is exp ose d relative to M if x is not the endp oint of a heavy edg e. An odd cycle C with V ( C ) = { x 0 , x 1 , ..., x 2 k } a nd E ( C ) = { x i x i +1 : 0 ≤ i ≤ 2 k − 1 } ∪ { x 2 k , x 0 } , such that x 1 x 2 , x 3 x 4 , ..., x 2 k − 1 x 2 k ∈ M is a blossom relative to M . The vertex x 0 is the b ase of the blossom. The stem is an even length alternating path joining the base of a blossom a nd an exp os ed vertex for M . The base is the only common vertex to the blossom and the s tem. A flower is a bloss om and its stem. A p osy cons ists of tw o (not necessarily disjoint) blossoms jo ined by an o dd length alternating path whose first and last edges b elong to M . The endp oints o f the path a re exactly the ba ses of the tw o blossoms. Theorem 1.3 [25] F or a gr aph G , the fol lowing pr op erties ar e e quivalent: (i) G is a K¨ onig-Egerv´ ary g r aph; (ii) t her e exist no flower and no p o s y re lative to some maximum matching M ; (iii) t her e exist no flower and no p o s y re lative to every maximum matching M . ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 1 2 2 k + 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 1 2 2 k ✇ ✇ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✇ ❅ ❅ ❅ v ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ v ✇ ✇ ✇ ✇ ✇ ✇ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❅ ❅ ❅    Figure 2: F orbidden configurations . The vertex v is not adjacent to the ma tching edg es (namely , dashed edges). In [9], Gavril defined the so-ca lled red/blue-split graphs, as a common genera lization of K-E a nd split graphs. Na mely , G is a r e d/blue-split gr aph if its edges can be c o lored in red and blue suc h that V ( G ) can be partitio ned into a red and a blue indep endent set (where r e d or blue indep ende n t set is a n indep endent set in the g raph made of red or blue edges). In [12], K o rach et al. des crib ed red/blue- s plit g raphs in terms of exc luded configuratio ns, which led them to the following characterization of K-E g r aphs. Theorem 1.4 [12] L et M b e a maximum matching in a gr aph G . Then G is a K¨ onig- Egerv´ ary gr aph if and only if G do es not c ontain one of the forbidden c onfigur ations, depicte d in Figur e 2, with r esp e ct to M . 3 In [21], Lo v as z gives a characterization of K-E graphs having a p er fect ma tching, in terms of c e rtain for bidden subgr aphs with r esp ect to a specific perfect matc hing of the graph. The problem of reco gnizing K-E gr a phs is p olynomia l as prov ed by Deming [5 ], of co mplexity O ( | V ( G ) | | E ( G ) | ). Gavril [9] has describ ed a r ecognition algo rithm for K-E g r aphs of co mplexity O ( | V ( G ) | + | E ( G ) | ). The proble m o f finding a ma ximum independent s et in a K-E gr aph is p o lynomial as proved by Deming [5]. The n umber d ( G ) = ma x {| S | − | N ( S ) | : S ∈ Ind( G ) } is called the critic al d iffer enc e of G . An indepe ndent s et A is critic al if | A | − | N ( A ) | = d ( G ), and the critic al indep endenc e numb er α c ( G ) is the cardinality o f a ma ximum critical independent set [26]. Clea rly , α c ( G ) ≤ α ( G ) holds for any gra ph G . It is known that the proble m of finding a critical indep endent set is po lynomially solv able [1, 26]. In [13] it w as shown that G is a K-E graph if and o nly if α c ( G ) = α ( G ), thus giving a p os itive ans wer to the Graffiti.pc 329 conjecture [4]. The deficiency of G , denoted by def ( G ), is defined a s the n um b er o f exp osed vertices relative to a maximum matching [22]. In other words, def ( G ) = | V ( G ) | − 2 µ ( G ). In [20] it was proven tha t the c r itical differ ence for a K-E graph G is given by d ( G ) = | c o re( G ) | − | N (core( G )) | = α ( G ) − µ ( G ) = def ( G ), and using this finding it was demonstrated that G is a K-E gr aph if and only if each o f its maxim um independent sets is critical. In this paper we g ive a new characterization of K-E graphs based on some commo n prop erty of its ma ximum matchings, and further we use it in or der to inv es tig ate K- E graphs in more detail. 2 Results Notice that a ll the ma ximum matchings of the g raphs G 1 and G 2 from Figure 3 are included in ( S, V ( G i ) − S ) , i = 1 , 2, for each S ∈ Ω( G i ) , i = 1 , 2 . O n the other hand, M 1 = { xu , y z } and M 2 = { xu, v z } are maximum matchings of the graph G 3 from Figur e 3, a nd S = { u, v } ∈ Ω( H 2 ), but M 1 * ( S, V ( G 3 ) − S ), while M 2 ⊆ ( S, V ( G 3 ) − S ). ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ❅ ❅ ❅ a b c x y G 1 ✇ ✇ ✇ ✇ ✇ ✇    G 2 ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ u v x z y G 3 Figure 3 : G 1 and G 2 are K¨ o nig–Ege r v´ ary g r aphs, but o nly in G 2 has a per fect matc hing . G 3 is not a K¨ o nig –Eger v´ ary gra ph. Theorem 2.1 F or a gr ap h G = ( V , E ) , the fol lowing pr op erties ar e e qu ivalent: (i) G is a K¨ onig-Egerv´ ary g r aph; (ii) e ach max imu m matching of G is c ontaine d in ( S, V − S ) for some S ∈ Ω( G ) ; (iii) e ach maximum matching o f G is c ontaine d in ( S, V − S ) for every S ∈ Ω( G ) . 4 Pro of. (i) = ⇒ (iii) Let G be a K-E gra ph. Supp ose that ther e exist some S ∈ Ω( G ) and a ma ximum matching M such that M * ( S, V − S ). According to Prop os ition 1.2, G can b e wr itten as G = S ∗ H and µ ( G ) = | V ( H ) | = | V − S | . Since S is indepe ndent and M * ( S, V − S ), there must be an edge in M ∩ E ( H ). Hence, we infer that µ ( G ) < | V ( H ) | , in con tr a diction with µ ( G ) = | V ( H ) | . Therefore, M must be contained in ( S, V − S ). (iii) = ⇒ (ii) It is clear. (ii) = ⇒ (i) Let S ∈ Ω( G ) enjoy the prop erty that eac h maximum matching o f G is contained in ( S, V − S ). Assume, on the contrary , that G is not a K-E graph, i.e., α ( G ) + µ ( G ) < | V ( G ) | . Let M = { a k b k : 1 ≤ k ≤ µ ( G ) } be a maximum matching in G . Since M ⊆ ( S, V − S ), we infer that µ ( G ) ≤ | S | = α ( G ), and one ma y supp ose that A = { a k : 1 ≤ k ≤ µ ( G ) } ⊆ S , while B = { b k : 1 ≤ k ≤ µ ( G ) } ⊆ V − S. In addition, it follows tha t µ ( G ) < | V − S | , b eca use | S | + | M | = α ( G ) + µ ( G ) < | V | = | S | + | V − S | = α ( G ) + | V − S | . Let x ∈ V − S − B and S x be the set of vertices v ∈ B such that there exis ts a path x = v 1 , v 2 , ..., v 2 k +1 = v , where v 2 i v 2 i +1 ∈ M , v 2 i ∈ A and v 2 i +1 ∈ B . W e sho w that the set S 1 = { x } ∪ S x ∪ ( S − M ( S x )) is indep endent, where M ( S x ) = { a j ∈ A : b j ∈ S x } . Claim 1 . { x } ∪ ( S − M ( S x )) is an independent set in G . Clearly , S − M ( S x ) is indep endent, as a subset o f S . In addition, if xy ∈ E , for some y ∈ S − M ( S x ), then, accor ding to the definition of S x , no edg e issuing from y belong s to M . Hence, M ∪ { xy } is a matching in G , la rger than M , in contradiction to the maximality of M . Therefore, { x } ∪ ( S − M ( S x )) is indep endent. Claim 2 . S x is independent. Otherwise, assume that b j b k ∈ E for some b j , b k ∈ S x . By definition of S x , there are t wo pa ths: P 1 : x = v 1 , v 2 , ..., v 2 p +1 = b j , where v 2 i v 2 i +1 ∈ M , v 2 i ∈ A and v 2 i +1 ∈ B , and P 2 : x = u 1 , u 2 , ..., u 2 q +1 = b k , where u 2 i u 2 i +1 ∈ M , u 2 i ∈ A and u 2 i +1 ∈ B . Case 1 . b k = v 2 s +1 is on the path P 1 (similarly , when b j = u 2 s +1 on the path P 2 ). Then, it follows that M 1 = { v 1 v 2 , v 3 v 4 , ..., v 2 s − 1 v 2 s } ∪ { v 2 s +3 v 2 s +4 , v 2 s +5 v 2 s +6 , ..., v 2 p − 1 v 2 p } ∪ { b j b k } is a matching with p edges, and M 2 = M 1 ∪ ( M − { v 2 i v 2 i +1 : 1 ≤ i ≤ p } ) is a maxim um matc hing in G . This con tra dicts the assumption that M 2 ⊆ ( S, V − S ), bec ause b j , b k ∈ S x ⊆ V − S . 5 Case 2 . The paths P 1 and P 2 hav e in common only the vertex x . The edge b j b k closes a cycle with he paths P 1 and P 2 . Now, the sets M 3 = { v 1 v 2 , v 3 v 4 , ..., v 2 p − 1 v 2 p } ∪ { u 3 u 4 , u 5 u 6 , ..., u 2 q − 1 u 2 q } ∪ { b j b k } , and M 4 = { v 2 i v 2 i +1 : 1 ≤ i ≤ p } ∪ { u 2 i u 2 i +1 : 1 ≤ i ≤ q } are disjoint matchings in G , b oth with p + q edges , while M 5 = M ∪ M 3 − M 4 is a maximum matching that satisfies M 5 * ( S, V − S ), in contradiction to the hypo thesis. Therefore, S x m ust be an indep endent set in G . Claim 3 . No edg e joins x to some vertex of S x . Suppo se, o n the con tr ary , that ther e is b j ∈ S x , such that xb j ∈ E . By the definition of S x , there is a pa th x = v 1 , v 2 , ..., v 2 p +1 = b j , where v 2 i v 2 i +1 ∈ M , v 2 i ∈ A and v 2 i +1 ∈ B . Then M 1 = { v 1 v 2 , v 3 v 4 , ..., v 2 p − 1 v 2 p } is a match ing in G with p edges, and M 2 = M ∪ M 1 ∪ { xb j } − { v 2 i v 2 i +1 : 1 ≤ i ≤ p } is a maxim um matching of G . Since x, b j ∈ V − S , it follows that M 2 * ( S, V − S ), again in con tr a diction to the hypothesis . Claim 4 . No edg e joins a vertex from S − M ( S x ) to a vertex of S x . Otherwise, as sume that there is y ∈ S − M ( S x ) , b j ∈ S x , such that xb j ∈ E . As ab ove, there is a path x = v 1 , v 2 , ..., v 2 p +1 = b j , where v 2 i v 2 i +1 ∈ M , v 2 i ∈ A and v 2 i +1 ∈ B . Then, the set M 1 = { v 1 v 2 , v 3 v 4 , ..., v 2 p − 1 v 2 p } is a match ing in G with p edges, and M 2 = M ∪ M 1 ∪ { y b j } − { v 2 i v 2 i +1 : 1 ≤ i ≤ p } is a matching o f G larg er than M , thus co ntradicting the maximality of M . Finally , we may c onclude that S 1 = { x } ∪ S x ∪ ( S − M ( S x )) is an indep endent set in G , but this le a ds to the follo wing inequality | S 1 | = | S | + 1 > α ( G ) , which clearly co ntradicts the f a ct that α ( G ) is the size of a maximum indep endent set in G . Prop ositi on 2. 2 If G = ( V , E ) is a K¨ onig-Egerv´ ary gr aph, then (i) for every maximum matching e ach exp ose d vertex b elongs to c or e ( G ) . (ii) at le ast one of the endp oints of every e dge of G is a µ -critic al vertex. Pro of. (i) By Theor em 2.1, ev er y ma ximum matching M is included in ( S, V − S ), for each maximum independent set S . Since | M | = | V − S | , we deduce that no exp osed vertex b elong s to V − S , and consequently , no exp osed vertex is in ∪{ V − S : S ∈ Ω( G ) } = V − ∩{ S : S ∈ Ω( G ) } = V − cor e( G ) . In other words, every exp osed v er tex b elong s to core( G ). 6 (ii) Suppose that uv ∈ E and v is not µ -critical, i.e., µ ( G − v ) = µ ( G ). If α ( G − v ) = α ( G ), then we get the following contradiction: | V | − 1 ≥ α ( G − v ) + µ ( G − v ) = α ( G ) + µ ( G ) = | V | . Therefore, we infer tha t α ( G − v ) = α ( G ) − 1, i.e., v ∈ cor e( G ). Hence, u ∈ N (cor e( G )), and, co ns equently , u is µ - critical, b ecaus e N (core( G )) is matched into core( G ) by every maximum matching in a K -E graph (by Theo rem 1 .1 (i) ). Remark 2.3 Th e c onverse of Pr op osition 2.2 (i) is fa lse (s e e the gr aphs in Figur e 4). ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅    W ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅    ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❅ ❅ ❅ H Figure 4: The non-K¨ onig -Eger v´ a ry gr a phs W and H hav e all exp o sed vertices in c or e ( W ) and cor e ( H ), resp ectively . Remark 2.4 Pr op osition 2.2 (ii) is not sp e cific for K- E gr aphs; se e, for instanc e, the gr aph G 1 fr om Figur e 5 . On the other hand, ther e exist gr aphs wher e the endp oints of (a) so m e e dges ar e not µ -critic a l (e.g., the e dge ab o f the gr aph G 2 fr om Fig ur e 5), (b) e ach e dge ar e not µ - critic al (e.g. , the gr aph G 3 fr om Figur e 5). ✇ ✇ ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅ G 1 ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅ a b G 2 ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ G 3 Figure 5: All G i , i = 1 , 2 , 3 , are no t K ¨ onig–Eg erv´ ary gr aphs. Prop ositi on 2. 5 L et G b e a K¨ onig-Egerv´ ary gr aph G and v ∈ V ( G ) b e such that G − v is stil l a K¨ onig-Egerv´ ary gr aph. Then v ∈ core( G ) if and only if ther e exists a m ax imu m matching that do es n ot satura t e v . Pro of. Since v ∈ core( G ), it follows that α ( G − v ) = α ( G ) − 1. Consequently , we have α ( G ) + µ ( G ) − 1 = | V ( G ) | − 1 = | V ( G − v ) | = α ( G − v ) + µ ( G − v ) which implies that µ ( G ) = µ ( G − v ). In other w ords , there is a maximum matc hing in G no t satur a ting v . Conv ersely , supp ose that ther e exists a maximum matching in G that do es not s atu- rate v . Since, by Theorem 1.1 (i) , N (co r e( G )) is ma tched in to co re( G ) by every maximum matching, it follows that v / ∈ N (cor e( G )). Assume that v / ∈ core ( G ). By Theo r em 1.1 (ii) , H = G − N [c o re( G )] is a K-E graph, H has a perfect matc hing and ev ery m aximum matching M o f G is of th e form M = M 1 ∪ M 2 , wher e M 1 matches N (cor e( G )) into core( G ), w hile M 2 is a p er fect matching of H . Consequently , v is sa turated by every ma ximum matching of G , in contradiction with the h y po thesis o n v . 7 Remark 2.6 Th e ab ov e pr op osition is not true if G − v is n ot a K - E gr aph; e.g., e ach maximum matching o f the gr aph G fr om F igur e 1 satur ates c ∈ c or e ( G ) = { a, b , c } . Corollary 2.7 F or every bip artite gr aph G , the vertex v ∈ core( G ) if and only if t her e exists a maximum matching that do es not satur ate v . 3 Conclusions In this paper we give a new c haracter ization of K¨ onig-E gerv´ ar y gra phs similar in form to Sterb oul’s Theorem 1.3. It seems to be interesting to characterize K¨ onig -Egerv ´ a ry graphs with unique maximum indep endent sets. References [1] A. A. Ageev, On finding critic al indep endent and vertex sets , SIAM J. Discrete Mathematics 7 (19 9 4) 293 –295 . [2] J. M. Bourjolly , P . L. Hammer, B. Simeone , No de weighte d gr aphs having K¨ onig- Egervary pr op erty , Math. Progra mming Study 22 (1984) 44-63 . [3] J. M. 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