On Duality between Local Maximum Stable Sets of a Graph and its Line-Graph

On Dualit y b et w een Lo cal Maxim um Stable Sets of a Graph and its Line-Graph V adim E. Levit Departmen t of Computer Science and Mathematics Ariel Univ ersit y Cen ter of Samaria, ISRAEL levitv@ariel.ac.il Eugen Mandrescu Departmen t of Computer Science Holon Institute of T ec hnology , ISRAEL eugen m@hit.ac.il Abstract G is a K¨ onig-Egerv´ ary gr aph pro vided α ( G ) + µ ( G ) = | V ( G ) | , where µ ( G ) is the size of a maximum matc hing and α ( G ) is th e cardinalit y of a maximum stable set, [3 ], [22]. S is a lo c al maxi m um stable set of G , and we write S ∈ Ψ( G ), if S is a maximum stable set of the subgraph induced by S ∪ N ( S ), where N ( S ) is the neighborho od of S , [12]. Nemhauser and T rotter Jr. prov ed that any S ∈ Ψ( G ) is a sub set of a maximum stable set of G , [20]. In this paper w e demonstrate that if S ∈ Ψ( G ), th e subgraph H induced b y S ∪ N ( S ) is a K¨ onig-Egerv´ ary graph, and M is a maximum matc hing in H , th en M is a local maximum stable set in the line graph of G . Keywords: Line graph, K¨ onig-Egerv´ ary graph, maximum matc hing, local maxi- mum stable set. 1 In tro duction Throughout this pap er G = ( V , E ) is a simple (i.e., a finite, undirected, lo opless and without m ultiple edges) graph with v ertex set V = V ( G ) and edge set E = E ( G ). If X ⊂ V , then G [ X ] is the subgr aph of G spanned by X . By G − W we mean the subgraph G [ V − W ], if W ⊂ V ( G ). W e also denote by G − F the partial subgraph of G obtained b y deleting the edges of F , for F ⊂ E ( G ), and we write shor tly G − e , whenever F = { e } . If A, B ⊂ V a re disjoint and non-empty , then by ( A, B ) we mean the se t { ab : ab ∈ E , a ∈ A, b ∈ B } . The neighb orho o d of a vertex v ∈ V is the set N ( v ) = { w : w ∈ V and v w ∈ E } . If | N ( v ) | = 1, then v is a p endant vertex . W e denote the neighb orho o d of A ⊂ V by N G ( A ) = { v ∈ V − A : N ( v ) ∩ A 6 = ∅ } a nd its close d neighb orho o d b y N G [ A ] = A ∪ N ( A ), or shortly , N ( A ) and N [ A ], if no ambiguity . K n , C n denote resp ectiv ely , the complete graph on n ≥ 1 vertices, and the chordless cycle on n ≥ 3 vertices. A g raph having no K 3 as a subgra ph is a triangle-fr e e gr aph . 1 A stable set in G is a s et of pairwise non-adjacent vertices. A s ta ble se t of ma xim um size will be r e fer red to as a maximum st abl e set of G , and the stability numb er o f G , denoted b y α ( G ), is the ca rdinalit y of a maximum stable set in G . In the sequel, by Ω( G ) we denote the set o f all maximum stable sets of the graph G . A set A ⊆ V ( G ) is a lo c al maximum stable set of G if A is a maxim um stable set in the subgraph spa nned b y N [ A ], i.e., A ∈ Ω ( G [ N [ A ]]), [12]. Let Ψ( G ) sta nd for the set o f all lo cal maximum stable sets of G . Clearly , every set S co nsisting of only p endan t vertices b elongs to Ψ( G ). Nevertheless, it is not a must for a lo cal maximum stable set to contain pendant v ertices . F or instance, { e, g } ∈ Ψ( G ), where G is the g raph from Figure 1. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c d g f e Figure 1: A g r aph having v a rious lo cal maximum stable sets. The following theorem concerning maximum stable sets in g e neral graphs, due to Nemhauser and T r otter Jr. [2 0], shows that some stable sets ca n b e enlar ged to maximum stable sets. Theorem 1.1 [20] Every lo c al maximum stable set of a gr aph is a subset of a maximum stable set. Let us no tice that the conv erse of Theorem 1 .1 is trivia lly true, b ecause Ω( G ) ⊆ Ψ( G ). The graph W from Figure 1 has the pro p erty that any S ∈ Ω( W ) contains some lo cal max im um stable set, but these lo cal maxim um stable sets ar e of different cardinalities: { a, d, f } ∈ Ω( W ) and { a } , { d, f } ∈ Ψ ( W ), while for { b, e, g } ∈ Ω( W ) only { e, g } ∈ Ψ( W ). How ever, there ex ists a graph G satisfying Ψ( G ) = Ω( G ), e.g., G = C n , for n ≥ 4. A matching in a graph G = ( V , E ) is a set o f edges M ⊆ E such that no tw o edges of M s hare a common vertex. A maximum matching is a ma tc hing of ma xim um size, deno ted by µ ( G ). A matching is p erfe ct if it satur ates all the v ertices o f the gr aph. A matching M = { a i b i : a i , b i ∈ V ( G ) , 1 ≤ i ≤ k } of a graph G is called a u niquely r estricte d matching if M is the unique p e rfect matching of G [ { a i , b i : 1 ≤ i ≤ k } ], [6]. Recently , a g eneralization of this conc e pt, namely , a sub gr aph r estricte d matching has b een studied in [5 ]. Kroghda l found that a matching M of a bipa r tite g r aph is uniquely r estricted if a nd only if M is alternating cycle-free (see [10]). This statement w as observed for g eneral gra phs by Golumbic et al . in [6]. In [12], [13], [16], [1 7], [18] w e show ed that, under certain conditions in volving uniquely restricted matchings, Ψ ( G ) for ms a gree do id on V ( G ). The classes of graphs, where greedoids were fo und include tr ees, bipartite gra phs, triangle- free gr aphs, and well-co vered gr a phs. Recall that G is a K¨ onig-Egerv´ ary gr aph provided α ( G ) + µ ( G ) = | V ( G ) | ([3], [22]). As a w ell-known example, any bipa rtite gr aph is a K¨ onig -Egerv´ ary gr a ph ([4], [9]). P roper ties of K¨ onig- Egerv´ a ry graphs w ere discussed in a n umber of papers, e.g., [1 ], [7], [8], [1 1], [1 4], [15], [19], [21]. Let us notice that if S is a stable set and M is a matching in a graph G s uch that | S | + | M | = | V ( G ) | , it follo ws that S ∈ Ω( G ) , M is a maximu m matching, and G is a K¨ o nig-Egerv´ ary graph, b ecause | S | + | M | ≤ α ( G ) + µ ( G ) ≤ | V ( G ) | is true for any graph. 2 The line g raph of a g r aph G = ( V , E ) is the graph L ( G ) = ( E , U ), where e i e j ∈ U if e i , e j hav e a co mmon endp oint in G . In this pa p er w e giv e a s ufficien t conditio n in terms of subgr aphs of G that ens ure that its line graph L ( G ) has proper loca l ma xim um sta ble sets. In other w ords, w e demo nstrate that if: S ∈ Ψ( G ), the subgraph H induced b y S ∪ N ( S ) is a K¨ o nig-Egerv´ ary graph, a nd M is a maximum matc hing in H , then M is a lo cal maximum stable set in the line graph o f G . It turns out that this is also a sufficient condition for a matching of G to be extenda ble to a maximum matching. 2 Maxim u m matc hings and lo cal maxim um stable sets In a K ¨ onig-Eg erv´ ary graph, maximum matchings hav e a spe cial pro p erty , emphasized b y the following statement . Lemma 2.1 [14] Every maximum matching M of a K¨ onig-Egerv´ ary gr aph G is c ontaine d in e ach ( S, V ( G ) − S ) and | M | = | V ( G ) − S | , wher e S ∈ Ω( G ) . F or example, M = { e 1 , e 2 , e 3 } is a maxim um matching in the K¨ onig-Eg e r v´ ary graph H (from Fig ure 2 ), S = { a, b, c, d } ∈ Ω( H ) and M ⊂ ( S, V ( H ) − S ). On the other hand, M 1 = { xz , y v } , M 2 = { y z , uv } are maximum matchings in the non-K ¨ onig-Eg erv´ ary graph G (depicted in Fig ur e 2), S = { x, y } ∈ Ω( G ) and M 1 ⊂ ( S, V ( G ) − S ), while M 2 * ( S, V ( G ) − S ). ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅    ❅ ❅ ❅ x y z v u G ✇ ✇ ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ e 1 e 2 e 3 a b c d H Figure 2: { x, y } ∈ Ω ( G ) and { a, b, c , d } ∈ Ω( H ). Clearly , (maximum ) matchings in a gra ph G corr espond to (maximum , resp e c tiv ely) stable sets in L ( G ) and vice versa. How ever, not every matching M in G gives birth to a lo cal maximum stable set in L ( G ), even if M can b e enlar ged to a maximum matching. ✇ ✇ ✇ ✇ ✇ ✇ ✇    e 1 e 2 e 3 e 5 e 7 e 4 e 6 x y z v G ✇ ✇ ✇ ✇ ✇ ✇ ✇       ❅ ❅ ❅ e 1 e 2 e 3 e 5 e 7 e 4 e 6 H Figure 3: The g raph G and its line-gr aph H = L ( G ). F or instance, M 1 = { e 1 , e 6 } , M 2 = { e 3 , e 6 } a re b oth ma tc hings in the graph G from Figure 3, but only M 1 is a lo cal ma xim um stable set in L ( G ). Remark that S 1 = { v , z } ∈ Ψ ( G ) , S 2 = { x, y } / ∈ Ψ( G ) and ea c h M i is a max im um matching in G [ N [ S i ]], for i ∈ { 1 , 2 } . 3 Theorem 2.2 If S ∈ Ψ( G ) , H = G [ N [ S ]] is a K¨ onig-Egerv´ ary gr aph, and M is a max imu m matching in H , then M is a lo c al maximum stable set in L ( G ) . Pro of. Let M = { e i = v i w i : 1 ≤ i ≤ µ ( H ) } . Accor ding to Lemma 2.1, it follows that M ⊆ ( S, V ( H ) − S ) a nd | M | = | V ( H ) − S | , bec ause H is a K ¨ onig-Eg erv´ ary gra ph. Consequently , without los s of g eneralit y , w e may suppo se that { v i : 1 ≤ i ≤ µ ( H ) } ⊆ S , while V ( H ) − S = { w i : 1 ≤ i ≤ µ ( H ) } . Since N H ( v i ) = N G ( v i ) ⊆ N ( S ) = V ( H ) − S , we hav e that N L ( G ) [ M ] = E ( H ) ∪ { e = w t ∈ E : w ∈ V ( H ) − S, t / ∈ S } . Hence, every e ∈ N L ( G ) [ M ] − V ( L ( H )) is incident in G to some w i . Assume that M is not a maximum stable set in L ( G ), i.e., ther e exists some stable set Q ⊆ N L ( G ) [ M ], suc h that | Q | > | M | . In other words, Q is a matching using edge s from E ( H ) ∪ { e = wt ∈ E : w ∈ V ( H ) − S, t / ∈ S } , larger tha n M . Let F = ( M − Q ) ∪ ( Q − M ). Since M a nd Q ar e matc hings, every vertex app earing in G [ F ] has at most o ne incident edg e from each of them, and the ma xim um degree of a vertex in G [ F ] is 2 . Hence, G [ F ] co nsists o f o nly disjoin ts c hor dless pa ths and cycles. Moreov er, every path and every cycle in G [ F ] alternates b et ween edges of Q a nd edges of M . Since | Q | > | M | , it follows that G [ F ] has a comp onent with mor e edges of Q than of M . Such a co mp onent c an only b e a path, say P x,y , that sta rts and ends with edges from Q (more precisely , from Q − M ) and and x, y are not sa turated by edges b elonging to M . Hence, P x,y m ust hav e an o dd num ber of edg e s . Case 1. P x,y contains only one edge, namely xy . This is not pos sible, since a t lea st one of the vertices x, y belo ng s to V ( H ) − S and is s aturated by M . Case 2. P x,y contains at least three edges. Let xa, by ∈ Q b e the fir st and the la st edg es on P x,y . Clea rly , E ( P x,y ) * E ( H ), b ecause, otherwise ( M − E ( P x,y )) ∪ ( E ( P x,y ) − M ) is a ma tc hing in H , larg e r than M , in contradiction with the maximality of M . Hence, P x,y contains edges from M , that a lternates with edge s fro m ( E ( H ) − M ) ∪ W , where W = { w t ∈ E ( G ) : w ∈ V ( H ) − S, t ∈ U } , with U = ( S − { v i : 1 ≤ i ≤ µ ( H ) } ) ∪ ( V ( G ) − V ( H ) } 6 = ∅ . Therefore, each second vertex on P x,y m ust b elong to V ( H ) − S . Consequently , we infer that als o y ∈ V ( H ) − S , and hence, it is sa turated b y M , a contradiction. Notice tha t M = { e 5 , e 7 } ∈ Ψ( L ( G )), while there is no S ∈ Ψ( G ), such that M is a maximum matching in G [ N [ S ]], where G is depicted in Figure 4. In other w ords, the conv erse of Theo rem 2.2 is not true. Clearly , every matc hing c a n b e enlarged to a maximal matc hing, which is not ne c essarily a max imum matching. F or instance, the gr aph G in Figure 5 do es no t contain any maximum matching including the matc hing M = { e 0 , e 1 , e 2 } . The following result sho ws that, under certain co nditio ns , a matching can be extended to a max imum matching. 4 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ e 1 e 4 e 3 e 2 e 5 e 6 e 7 e 8 G ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ❅ ❅ ❅ e 1 e 4 e 3 e 2 e 5 e 6 e 7 e 8 L ( G ) Figure 4: M = { e 5 , e 7 } is a matching in G and lo cal maximum sta ble set in L ( G ). ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ e 0 e 1 e 2 G Figure 5: { e 0 , e 1 , e 2 } is a maximal but not a maximum matching. Corollary 2.3 If S ∈ Ψ( G ) , H = G [ N [ S ]] is a K¨ onig-Egerv´ ary gr aph, and M is a maximum matching in H , then ther e exists a maximu m matching M 0 in G such that M ⊆ M 0 . Pro of. Accor ding to Theor em 2.2, M is a lo cal maximum s ta ble set in L ( G ). By Theo rem 1.1, there is some M 0 ∈ Ω( L ( G )), such that M ⊆ M 0 . Hence, M 0 is a maximum matching in G containing M . Let us notice tha t Corolla ry 2.3 c an not b e genera lized to any subgr aph of a non-bipar tite K¨ o nig-Egerv´ ary graph. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c d e f g h i Figure 6: M = { ab, c d, f h } is a maximum matching in N [ { a, c, f } ]. F or instance, the graph G depicted in Figure 6 is a K¨ onig-Eger v´ ary g raph, S = { a, c, f } ∈ Ψ( G ), and M = { ab, cd, f h } is a max imum matching in G [ N [ S ]], which is not a K¨ onig- Egerv´ ary graph, but ther e is no maximum matching in G that includes M . Since any subgraph o f a bipartite graph is also bipartite, w e obta in the following result. Corollary 2.4 If G is a bip artite gr aph, S ∈ Ψ( G ) and M is a maximum matching in G [ N [ S ]] , t hen t her e exists a maximum matching M 0 in G such t ha t M ⊆ M 0 . 3 Conclusions W e show ed tha t there is some connection betw een Ψ( G ) and Ψ ( L ( G )). Let us notice that there ar e graphs whose line graphs hav e no prop er lo cal ma xim um stable sets (see the graphs in Figure 7). Moreov er, there ar e gra phs whose iterated line gr a phs have no prop er lo cal ma xim um stable set, e.g., each C n , for n ≥ 3, since C n and L ( C n ) are is o morphic. 5 ✇ ✇ ✇ ✇ ✇ ✇ e 1 e 7 e 2 e 5 e 6 e 3 e 4 G ✇ ✇ ✇ ✇ ✇ ✇ ✇       ❅ ❅ ❅ ❅ ❅ ❅    ❅ ❅ ❅    ❅ ❅ ❅ e 1 e 2 e 3 e 4 e 5 e 6 e 7 L ( G ) Figure 7: Both G and its line g raph L ( G ) hav e no lo cal maximum stable sets. An interesting op en question rea ds as follows. 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