On the Intersection of All Critical Sets of a Unicyclic Graph

On the In tersection of All Critical Sets of a Unicyclic Graph V adim E. Levit Ariel Univ ersit y Cen ter of Samaria, Israel levitv@ariel.ac.il Eugen Mandrescu Holon Institute of T ec hnology , Israel eugen m@hit.ac.il Abstract A set S ⊆ V is indep endent in a graph G = ( V , E ) if no tw o vertices from S are adj acent. The indep endenc e numb er α ( G ) is th e cardinality of a max- im um ind ependent set, while µ ( G ) is the size of a maximum matc hing in G . If α ( G ) + µ ( G ) = | V | , th en G is called a K¨ onig-Egerv´ ary gr aph . The number d c ( G ) = max {| A | − | N ( A ) | : A ⊆ V } is called the critic al di ffer enc e of G [21 ], where N ( A ) = { v : v ∈ V , N ( v ) ∩ A 6 = ∅} . By core( G ) (corona( G )) w e denote the inters ection (union, respectively) of all maximum independ en t sets, while b y ker ( G ) w e mean the in tersection of all critical indep endent sets . A connected graph ha ving only one cycle is called unicyclic . It is k no wn that the relation ker ( G ) ⊆ core( G ) h olds for every graph G [13 ], while the equality is true for bipartite graphs [14]. F or K¨ onig-Egerv´ ary un icyclic graphs, the difference | core( G ) | − | k er ( G ) | may equal any non-negative integer. In this pap er w e prov e that if G is a non- K ¨ onig-Egerv´ ary unicyclic graph, then: (i) ker ( G ) = core( G ) and (ii) | corona( G ) | + | core( G ) | = 2 α ( G ) + 1. P ay attention that | corona( G ) | + | core ( G ) | = 2 α ( G ) holds for every K¨ onig-Egerv´ ary graph [14]. Keywords: max imum indep endent set, core, corona, matching, critical set, unicyclic graph, K¨ onig-Egerv´ ary graph. 1 In tro du ction Throughout this pape r G = ( V , E ) is a simple (i.e., a finite, undir ected, lo opless a nd without m ultiple edges) g raph with vertex set V = V ( G ) and edge set E = E ( 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 de no te the pa rtial subgraph of G obtained by deleting the edges of F , a nd we use G − e , if W = { e } . If A, B ⊂ V and A ∩ B = ∅ , then ( A, B ) stands fo r the s et { e = ab : a ∈ A, b ∈ B , e ∈ E } . The neighborho o d of a vertex v ∈ V is the set N ( v ) = { w : w ∈ V and v w ∈ E } , and 1 N ( A ) = ∪{ N ( v ) : v ∈ A } , N [ A ] = A ∪ N ( A ) fo r A ⊂ V . By C n , K n we mean the chordless cycle on n ≥ 4 vertices, and resp ectiv ely the complete graph on n ≥ 1 vertices. A set S of vertices is indep endent if no tw o vertices from S a r e adjace n t, and a n independent set o f maximum size will b e referr ed to as a maximu m indep endent set . The indep endenc e numb er o f G , denoted b y α ( G ), is the cardinality o f a maximu m independent set of G . Let co re( G ) = ∩{ S : S ∈ Ω( G ) } [9], a nd coro na( G ) = ∪{ S : S ∈ Ω( G ) } [2], where Ω( G ) = { S : S is a maximum indep endent set of G } . Theorem 1. 1 [2] F or every S ∈ Ω ( G ) , t her e is a matching fr om S − c or e ( G ) into c or ona ( G ) − S . An edge e ∈ E ( G ) is α - critic al whenever α ( G − e ) > α ( G ). Notice that α ( G ) ≤ α ( G − e ) ≤ α ( G ) + 1 holds for each edge e . The num b er d ( X ) = | X | − | N ( X ) | , X ⊆ V ( G ), is called the differ enc e of the set X , while d c ( G ) = max { d ( X ) : X ⊆ V } is called the critic al differ enc e of G . A set U ⊆ V ( G ) is critic al if d ( U ) = d c ( G ) [2 1]. The num b er id c ( G ) = ma x { d ( I ) : I ∈ Ind( G ) } is calle d the critic al indep endenc e differ enc e o f G . I f A ⊆ V ( G ) is indep endent and d ( A ) = id c ( G ), then A is called a critic al indep endent set [21]. C le a rly , d c ( G ) ≥ id c ( G ) is true for every graph G . Theorem 1. 2 [2 1] The e quality d c ( G ) = id c ( G ) holds for every gr aph G . F or a gr a ph G , let denote ker( G ) = ∩ { S : S is a critic al indep endent set } . Theorem 1. 3 If G is a gr aph, then (i) [13] ker ( G ) is a critic al indep endent s et and ker ( G ) ⊆ cor e( G ) ; (ii) [14] ker ( G ) = core( G ) , whenever G is bip artite. A matching (i.e., a set of no n-inciden t edg es o f G ) o f ma xim um ca rdinalit y µ ( G ) is a maximum matching , and a p erfe ct matching is one covering all vertices of G . An edge e ∈ E ( G ) is µ - critic al provided µ ( G − e ) < µ ( G ). It is w ell-known that ⌊ n/ 2 ⌋ + 1 ≤ α ( G ) + µ ( G ) ≤ n hold for any gr aph G with n vertices. If α ( G ) + µ ( G ) = n , then G is called a K¨ onig -Egerv´ ary gr aph [3], [1 8]. Several prop erties of K¨ onig-Eger v ´ ary gra phs a re presented in [8], [10], [12]. According to a celebrated result of K¨ onig, [7], and Eger v´ ary , [5], any bipartite gr aph is a K¨ o nig-Egerv´ ary g raph. This class includes also non-bipa r tite gr aphs (see, for insta nce, the graph G in Figure 1). ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c u v x y G Figure 1: A K¨ o nig-Egerv´ ary graph with α ( G ) = |{ a, b, c, x }| and µ ( G ) = |{ au, cv , xy }| . 2 Theorem 1. 4 [1 0] If G is a K¨ onig-Egerv´ ary gr aph, then every maximum matching matches N ( c or e ( G )) int o c or e ( G ) . The gra ph G is c a lled un icyc lic if it is connected and has a unique cycle, which w e denote by C = ( V ( C ) , E ( C )). L e t N 1 ( C ) = { v : v ∈ V ( G ) − V ( C ) , N ( v ) ∩ V ( C ) 6 = ∅ } , and T x = ( V x , E x ) be the tree of G − xy containing x , where x ∈ N 1 ( C ) , y ∈ V ( C ). ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ❅ ❅ ❅ u v x y w c a b d t G ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ u v x a b T x Figure 2: G is a unicyc lic non-K¨ onig-Egerv´ ary graph with V ( C ) = { y , d, t, c, w } . The following result shows that a unicyclic graph is either a K¨ o nig -Egerv´ ary gr aph or each edge of its cycle is α -cr itical. Lemma 1.5 [16] If G is a unicyclic gr aph of or der n , then (i) n − 1 ≤ α ( G ) + µ ( G ) ≤ n ; (ii) n − 1 = α ( G ) + µ ( G ) if and only if e ach e dge of the unique cycle is α -critic al. Theorem 1. 6 [1 6] L et G b e a unicyclic non-K ¨ onig-Egerv ´ ary gr aph. Then the fol lowing assertions ar e true: (i) e ach W ∈ Ω ( T x ) c an b e enlar ge d to some S ∈ Ω ( G ) ; (ii) S ∩ V ( T x ) ∈ Ω ( T x ) for every S ∈ Ω ( G ) ; (iii) core ( G ) = ∪ { core ( T x ) : x ∈ N 1 ( C ) } . Unicyclic gra phs k eep enjoying plent y of interest, as o ne can see , for instance, in [1], [4], [6 ], [11], [17], [19], [20]. In this pa p er we analyze the relationship b et ween several parameters of a unicyclic graph G , na mely , core( G ), corona( G ), ker ( G ). 2 Results Lemma 2.1 If G is a unicyclic non-K¨ onig-Egerv´ ary gr aph, then (i) c or e ( G ) ∩ N [ V ( C )] = ∅ ; (ii) ther e exists a matching fr om N (core( G )) into cor e ( G ) . Pro of. (i) Let ab ∈ E ( C ). By Lemma 1.5 (ii) , the edge ab is α -critical. Hence ther e are S a , S b ∈ Ω ( G ), such that a ∈ S a and b ∈ S b . Since a / ∈ S b , it fo llows that a / ∈ core( G ), and b ecause a ∈ S a , we infer that N ( a ) ∩ co re( G ) = ∅ . Conseq uen tly , we obtain that core( G ) ∩ N [ V ( C ) ] = ∅ . (ii) If cor e( G ) = ∅ , then the conclusion is clear. 3 Assume that cor e( G ) 6 = ∅ . By T he o rem 1.4, in each tree T x there is a matching M x from N (core( T x )) into core( T x ). By part (i) , w e have that V ( C ) ∩ N [co re ( G )] = ∅ . T aking in to acco unt Theorem 1.6 (ii) , we see that the union of all these matchings M x gives a matching fro m N (core( G )) into core( G ). It is worth mentioning that the ass e r tion in Lemma 2 .1 (ii ) is true for every K¨ onig- Egerv´ ary gr aph, by Theo rem 1.4. The graph G 2 from Figure 3 shows that Lemma 2.1 (i) may fail for unicyclic K¨ onig -Egerv´ ary graphs. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇    x y z G 2 Figure 3: K¨ onig-Eger v´ ary gr aphs with core ( G 1 ) = { a, b, c } and core( G 2 ) = { x, y , z } . Theorem 2. 2 If G is a K¨ onig-Egerv´ ary gr aph, t hen (i) [10] N (cor e ( G )) = ∩{ V ( G ) − S : S ∈ ( G ) } , i.e., N (core( G )) = V ( G ) − coro na ( G ) ; (ii) [14] | c orona( G ) | + | core( G ) | = 2 α ( G ) . ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ❅ ❅ ❅ a b c G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ x y G 2 Figure 4: Non-K ¨ onig-Eg erv´ ary g raphs with co r e ( G 1 ) = { a, b } a nd cor e ( G 2 ) = { x, y } . The graphs G 1 , G 2 from Figure 4 satisfy corona( G 1 ) ∪ N (cor e( G 1 )) = V ( G 1 ) − { c } , while corona( G 2 ) ∪ N (core( G 2 )) = V ( G 2 ). Theorem 2. 3 If G is u nicycli c non-K¨ onig -Egerv´ ary gr aph, then corona( G ) ∪ N (core( G )) = V ( G ) , corona( G ) = V ( C ) ∪ ( ∪ { cor ona ( T x ) : x ∈ N 1 ( C ) } ) . Pro of. F or the first equality , it is enough to show that V ( G ) ⊆ corona ( G ) ∪ N (core ( G )). Let a ∈ V ( G ) . Case 1 . a ∈ V ( C ). If b ∈ N ( a ) ∩ V ( C ), then, by Lemma 1 .5 (ii) , the edge ab is α -critical. Hence a ∈ corona ( G ). Case 2 . a ∈ V ( G ) − V ( C ). Since V ( G ) = V ( C ) ∪ ( ∪ { V ( T x ) : x ∈ N 1 ( C ) } ), it follows that there is some y ∈ N 1 ( C ) , such that a ∈ V ( T y ). According to Theor e m 2.2 (i) , we k no w that V ( T y ) = corona( T y ) ∪ N (core( T y )). By Theorem 1 .6 (i) , corona ( T x ) ⊆ co rona ( G ) for every x ∈ N 1 ( C ) . Therefore, either a ∈ cor o na ( T y ) ⊆ co rona ( G ), or a ∈ N (core ( T y )) ⊆ N ( core ( G )), beca use core ( T y ) ⊆ core( G ) in a ccordance with Theor em 1.6 (iii) . 4 Consequently , a ∈ co rona ( G ) ∪ N (cor e ( G ) ). In other words, we get that V ( G ) ⊆ co rona ( G ) ∪ N (co re ( G )) , as required. As for the s econd equality , let us notice that V ( C ) ⊆ corona ( G ), b y Case 1 . If a ∈ coro na ( G ) − V ( C ), then by Theorem 1 .6 (ii) , ther e ar e S ∈ Ω ( G ) and b ∈ N 1 ( C ) , such tha t a ∈ S ∩ V ( T b ) ∈ Ω ( T b ). Hence a ∈ coro na ( T x ), and ther efore, corona( G ) − V ( C ) ⊆ ∪ { corona ( T x ) : x ∈ N 1 ( C ) } . Theorem 1 .6 (i) as sures that ∪ { corona ( T x ) : x ∈ N 1 ( C ) } ⊆ cor ona ( G ) . In conclusio n, corona( G ) = V ( C ) ∪ ( ∪ { corona ( T x ) : x ∈ N 1 ( C ) } ). The g raph G 2 from Figure 4 shows that the equality | co r ona( G ) | + | core( G ) | = 2 α ( G ) is no t tr ue for unicyclic non-K ¨ onig-Eg erv´ ary g raphs. Theorem 2. 4 If G is a un icyc lic gr aph, then 2 α ( G ) ≤ | co rona( G ) | + | core( G ) | ≤ 2 α ( G ) + 1 . Mor e over, G is a non-K¨ onig-Egerv´ ary gr aph if and only if | corona( G ) | + | cor e( G ) | = 2 α ( G ) + 1 . Pro of. By The o rem 2.2 (ii) , the equality 2 α ( G ) = | cor o na ( G ) | + | core( G ) | holds for every unicy clic K¨ onig-Eger v´ ary gr aph G . Assume now that G is not a K¨ onig-Egerv´ ary graph. Let S ∈ Ω ( G ). According to Theorem 2.3 a nd Lemma 1.5 (i) , we infer that | S | + | cor ona( G ) − S | + | N (core ( G )) | = | V ( G ) | = α ( G ) + µ ( G ) + 1 , which implies | corona( G ) − S | + | N ( core ( G )) | = µ ( G ) + 1 . By Theorem 1.1, there is a matc hing M 1 from S − core ( G ) into coro na( G ) − S , which implies | S − core ( G ) | ≤ | co rona( G ) − S | . Lemma 1 .5 implies that ther e is a ma tc hing M from N (cor e ( G )) into cor e ( G ) , that can b e enlarg ed to a max im um matching, say M 2 , of G . Since M 2 matches µ ( G ) v ertices from A = (co rona( G ) − S ) ∪ ( N (core ( G ))) by means of µ ( G ) edges, and | A | = µ ( G ) + 1, it follows that M 2 − M ma tc hes | (coro na ( G ) − S ) | − 1 vertices from A into S − cor e ( G ), b ecause M satur a tes N (core ( G )) and no edge joins a vertex of co re ( G ) to some vertex fro m corona( G ) − S . Hence, tak ing into acco un t that M ∪ M 1 is a matching of G , while M 2 is a maximum ma tc hing , we obtain µ ( G ) = | M 2 | = | N ( core ( G )) | + | corona( G ) − S | − 1 ≤ ≤ | N (core ( G )) | + | S − co re ( G ) | = | M | + | M 1 | ≤ µ ( G ) , which implies | S − cor e ( G ) | = | co rona( G ) − S | − 1. Finally , we infer that | cor ona ( G ) | = | S ∪ (corona( G ) − S ) | = α ( G ) + | corona( G ) − S | = = α ( G ) + | S − c ore ( G ) | + 1 = 2 α ( G ) − | core ( G ) | + 1 , and this c o mpletes the pro of. 5 Theorem 2. 5 If G is a un icyc lic non-K¨ onig-Egerv´ ary gr aph, t hen ker ( G ) = ∪ { ker ( T x ) : x ∈ N 1 ( C ) } = co re( G ) . Pro of. Since T x is bipartite, b y Theor em 1.3 (ii) implies that ker ( T x ) = core( T x ), for every x ∈ N 1 ( C ) . According to Theorems 1.3 (i) a nd 1.6 ( iii ) , it follows that ker ( G ) ⊆ c ore( G ) = ∪ { cor e ( T x ) : x ∈ N 1 ( C ) } . Hence A x = ker ( G ) ∩ V ( T x ) ⊆ co re( G ) ∩ V ( T x ) = co re ( T x ) = ker ( T x ), fo r every x ∈ N 1 ( C ). Assume that A q 6 = k er ( T q ) for so me q ∈ N 1 ( C ). It follows that d ( A q ) < d (k er ( T q )). Since, by Theore m 1.6 (i) , we hav e ker ( G ) ⊆ co re( G ) , Le mma 1.5 (ii) ens ur es that N [ V ( C ) ] ∩ ker ( G ) = ∅ . Consequently , the se t W = ( k er ( G ) − A q ) ∪ ker ( T q ) is inde- pendent, and sa tisfies d (ker ( G )) = d ( ∪ { ker ( G ) ∩ V ( T x ) : x ∈ N 1 ( C ) } ) = = X x ∈ N 1 ( C ) d (ker ( G ) ∩ V ( T x )) = X x ∈ N 1 ( C ) d ( A x ) = d ( A q ) + X x ∈ N 1 ( C ) −{ q } d ( A x ) < < d (ker ( T q )) + X x ∈ N 1 ( C ) −{ q } d ( A x ) = d ( W ) ≤ max { d ( X ) : X ⊆ V ( G ) } = d (ker ( G )) , which is a co n tr a diction. Therefore, we infer that ker ( G ) ∩ V ( T x ) = core( G ) ∩ V ( T x ) = core ( T x ) = k er ( T x ) hold for ea c h x ∈ N 1 ( C ) . Hence, ker ( G ) = cor e( G ) = ∪ { core ( T x ) : x ∈ N 1 ( C ) } = ∪ { k e r ( T x ) : x ∈ N 1 ( C ) } , as claimed. Remark 2 .6 If G is a un icycl ic K¨ onig-Egerv´ ary gr aph that is non-bip artite, t hen the differ enc e b etwe en | core( G ) | and | ker ( G ) | may e qual any non- ne gative inte ger. F or in- stanc e, the gr aph G 2 k +1 fr om Figur e 5 satisfies α ( G 2 k +1 ) = k + 3 , µ ( G 2 k +1 ) = k + 1 , while | c o re( G 2 k +1 ) | − | ker ( G 2 k +1 ) | = k − 1 , k ≥ 1 . ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ r r r r r ✇ ✇ ❅ ❅ ❅ G 2 k +1 x y z v 1 v 2 v 3 v 4 v 5 v 2 k v 2 k +1 Figure 5: ker ( G 2 k +1 ) = { x, z } , while core( G 2 k +1 ) = { x, z , v 1 , v 3 , ..., v 2 k − 1 } . 6 3 Conclusions The equality cor e ( G ) = ker ( G ) may fail for some non-bipar tite unicyclic K¨ onig-Eger v´ ary graphs; e.g., the gr aphs G 1 and G 2 from Figure 3 sa tisfy ker ( G 1 ) = { a, b } 6 = c o re( G 1 ) = { a, b, c } , while ker ( G 2 ) = core( G 2 ) = { x, y , z } . Problem 3 .1 C har acterize non-bip artite unicyclic K¨ onig-Egerv´ ary gr aphs G satisfying core( G ) = k er ( G ) . The non-unicyclic g raphs G 1 and G 2 from Figure 6 satisfy | co rona( G 1 ) | + | cor e( G 1 ) | = 2 α ( G 1 ) and | coro na ( G 2 ) | + | core( G 2 ) | = 2 α ( G 2 ) + 1. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ G 2 Figure 6: core( G 1 ) = { a, b, c } and core( G 2 ) = ∅ . Problem 3 .2 C har acterize gr aphs satisfying 2 α ( G ) ≤ | co rona( G ) | + | core( G ) | ≤ 2 α ( G ) + 1 . References [1] F. Belardo, M. Li, M. E nzo, S. K . Simi ´ c, J. W a ng, On the sp e ctr al r adius of unicyclic gr aphs with pr escrib e d de gr e e se quenc e , Linear Algebr a and its Applications 432 (2010) 232 3-2334. [2] E. B oros, M.C. Golumbic, V. E. Levit, On the nu mb er of vert ic es b elonging t o al l maximum stable sets of a gr aph , Discr ete Applied Mathematics 124 (20 02) 17-25 . [3] R. W. 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Discr ete Mathematics 3 (199 0) 4 31-438. 8