Well-Covered Trees

Well-Covered Trees Vadim E. Levit and Eugen Mandrescu Department of Computer Science Center for Technological Education Affiliated with Tel-Aviv University 52 Golomb Str., P.O. Box 305 Holon 58102, ISRAEL {levitv, eugen_m}@barley.cteh.ac.il Abstract. A graph is well-covered if every maximal stable set is also maximum. One theorem of Ravindra 16] asserts that a tree of order at least two is well-covered if and only if it has a perfect matching consisting only of pendant edges. The edge-join of two connected graphs G1 G2 is the graph G1 G2 obtained by adding an edge joining two vertices belonging to G1 G2 , respectively. If both newly adjacent vertices are of degree
2 in G1 G2 , respectively, then G1 G2 is an internal edge-join of G1 G2 . In this paper we show that any well-covered tree, having at least three vertices, can be recursively constructed using the internal edge-join operation. We also characterize well-covered trees in terms of distances between their vertices and their pendant vertices. 1. Introduction Throughout this paper G = (V E ) is a simple (i.e., a nite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E (G): If X V , then GX ] is the subgraph of G spanned by X . By G ; W we mean the subgraph GV ; W ] , if W V (G). We also denote by G ; F the partial subgraph of G obtained by deleting the edges of F , for F E (G), and we use G ; e, if W = feg. A stable set of maximum size will be referred as to a stability system of G, and the stability number of G, denoted by (G), is the cardinality of a stability system in G, 1]. Let (G) stand for the set fS : S is a stability system of Gg. The neighborhood of v 2 V is the set N (v) = fw : w 2 V and vw 2 E g. If GN (v)] is a complete subgraph in G, then v is a simplicial vertex of G. If every vertex of G is either simplicial or adjacent to a simplicial vertex, then G is a simplicial graph, 4]. For A V , we denote N (A) = fv 2 V ; A : N (v) \ A 6= g. If jN (v)j = jfugj = 1, then v is a pendant vertex and e = uv is a pendant edge of G. Let pend(G) be the set fv 2 V (G) : v is a pendant vertex in Gg. Let d(u v) stand for the distance between the vertices u v of G, and for any u 2 V (G) we dene the set D(u G) = fd(u v) : v 2 pend(G) ; fugg, and D(pend(G)) = fD(u G) : u 2 pend(G)g. If no ambiguity, we shall 1 Well-Covered Trees 2 write D(u) instead of D(u G). If k1 k2 are positive integers, by k1 k2] we mean the set fk 2 N : k1  k  k2g. Let Kn Pn designate, respectively, the complete graph on n  1 vertices and the chordless path on n  2 vertices. As usual, a tree is an acyclic connected graph. A tree having at most one vertex of degree  3 is called a spider, 10], or an aster, 8]. A graph G is well-covered if it has no isolated vertices and if every maximal stable set of G is also a maximum stable set, i.e., it is in (G). G is called very well-covered, 5], provided G is well-covered and jV (G)j = 2(G). Well-covered graphs were dened by Plummer in 1970, 13]. A number of classes of well-covered graphs were completely described e.g., wellcovered bipartite graphs (Ravindra 1977, 16]), very well-covered graphs (Favaron 1982, 5], and Staples 1975, 18]), well-covered block graphs and unicyclic graphs (Topp and Volkmann 1990, 20]), well-covered graphs of girth  6 (Finbow, Hartnell and Nowakowski 1993, 6]), well-covered cubic graphs (Campbell, Ellingham and Royle 1993, 2]), strongly well-covered graphs (Pinter 1994, 12]), well-covered graphs that contain neither 4- nor 5-cycles (Finbow, Hartnell and Nowakowski 1994, 7]), 4-connected clawfree well-covered graphs (Hartnell and Plummer 1996, 9]), well-covered simplicial, chordal, and circular arc graphs (Prisner, Topp and Vestergaard 1996, 15]), well-covered Koenig-Egervary graphs (Levit and Mandrescu 1998, 11]). An excellent survey on this subject is 14], due to Plummer 1993. Several new structural and recursive characterizations of dierent subclasses of well-covered graphs can be found in 17] (Sankaranarayana and Stewart, 1996). For an application of a simple structural characterization of well-covered graphs to their recognition problem see 19] (Tankus and Tarsi, 1997). It is easy to prove that: Any graph having a perfect matching consisting of pendant edges is well-covered, (in fact, it is very well-covered). Proposition 1. The converse is not generally true (see, e.g., the graphs in Figure 1). Moreover, there are well-covered graphs without perfect matchings, (for instance, K3 ). Nevertheless, following Favaron's characterization for very well-covered graphs (i.e., Theorem 1), one can assert that having a perfect matching is a necessary condition for very well-coveredness. v v v v v v v v v v Figure 1: Very well-covered graphs with no perfect mathching consisting of only pendant edges. Well-Covered Trees 3 Theorem 1. 5] For a graph G without isolated vertices the following are equivalent: (i ) G is very well-covered (ii ) there exists a perfect matching in G that satis es property P (iii ) there exists at least one perfect matching in G and every perfect matching in G satis es property P . Property P in the above theorem is dened as follows. Property P . A matching M in a graph G satises property P if for every edge xy 2 M , N (x) \ N (y) = , and N (x) ; fyg is adjacent to all of N (y) ; fxg. However, the converse of Proposition 1 is true for trees, as Ravindra has proved in 16], and also for graphs of girth greater than 5 and nonisomorphic to C7 and K1 , as Finbow, Hartnell and Nowakowski have proved in 6]. Since we mainly deal in the sequel, with well-covered trees, let recall the following Ravindra's result. Theorem 2. 16] For a tree T with at least three vertices, the following assertions are equivalent (i ) T is well-covered (ii ) T has a perfect matching consisting of pendant edges (iii ) T is very well-covered (iv ) T has exactly jV (T )j =2 pendant vertices. In this paper we show that any well-covered tree, having at least three vertices, can be recursively constructed, by means of a graph operation, that we call internal edge-join , and a family of trees, irreducible with respect to this operation. We also give a description of well-covered trees in terms of distances between their vertices and their pendant vertices. 2. Recursive structure of well-covered trees Denition 1. The edge-join of two connected graphs G1 G2 is the graph obtained by adding an edge joining two vertices belonging to G1 G2, respectively. If both of the newly adjacent vertices are of degree  2 in G1 G2 , respectively, then G1 G2 is an internal edge-join of G1 G2. The graph G is irreducible with respect to the internal edge-join operation if it cannot be expressed as an internal edge-join of two graphs. Clearly, any connected graph having a bridge can be written as an edgejoin of two graphs. Notice that the edge-join of two trees is a tree, and also that two trees can be internal edge-joined provided each one is of order  3. Using the edge-join operation, we recall the following well-known recursive denition of trees: G1 G2 4 Well-Covered Trees Denition 2. 1 is a tree. If is a tree, then 1 is a tree. Lemma 1. The well-covered spiders are well-covered trees irreducible with K T T K respect to internal edge-join operation. Proof. It is clear, because a spider contains at most one vertex of degree  3. v v v v v v v v v v v v v HH@@H v ;;v @H;v v vH v v v Figure 2: Well-covered spiders. Proposition 2. Any internal edge-join of two well-covered trees is a well- covered tree. Proof. Let T = T1 T2 be the internal edge-join of two well-covered trees T1 T2. These trees are not isomorphic to K1 or K2 , because they cannot be internal edge-joined. According to Theorem 2, there exist M1 M2 perfect matchings in T1 T2 , respectively, consisting of pendant edges. Since the edge-join was internal, M1  M2 is a perfect matching consisting of pendant edges in T . Consequently, by Theorem 2, T itself is well-covered. Proposition 3. If T is a well-covered tree with at least three vertices, then the graph T K2 , obtained by joining one non-pendant vertex of T to any vertex of K2 with an edge, is a well-covered tree. Proof. Let K2 = (fa bg fabg) and M be the perfect matching of T , consisting of only pendant edges, which exists by Theorem 2. Hence, T K2 is a well-covered tree according to Theorem 2, since M  fabg is a perfect matching consisting of pendant edges in T K2 . Using a similar reasoning, we show that the converse is also true, namely: Proposition 4. Any well-covered tree T non-isomorphic to K1 K2, contains at least one edge e connecting two non-pendant vertices, such that T ; e = T  K2 T = T K2 and T is a well-covered tree. Proof. Since V (T ) is nite and no two pendant vertices have a common neighbor, there exists some pendant vertex a 2 V (T ) whose neighbor b has degree 2. Then, by Theorem 2, T = T ; fa bg is a well-covered tree, as having a perfect matching consisting of pendant edges, and clearly T = T K2 . 0 0 0 0 0 Well-Covered Trees 5 Lemma 2. If T is a well-covered tree and A = fv : v 2 V (T ) deg(v)  3g is non-empty, then the subgraph of T spanned by A is a subtree of T . Proof. Suppose, on the contrary, that W = T A] is not a subtree of T , i.e., W is disconnected. Let w1 w2 2 A be in dierent connected components of W . Then the unique path P in T connecting w1 w2 uses at least one vertex x 2 V (T ) ; A. Hence, x has two neighbors on P and also a pendant neighbor, i.e., deg(x)  3, contradicting the fact that x 2 V (T ); A. Therefore, W must be connected, and since it has no cycles, it is a subtree of T . Proposition 5. For any well-covered tree T having at least two vertices of degree  3, there exists some edge e such that T ; e = T1  T2 T = T1 T2, T1 and T2 are well-covered trees, and pend(T ) = pend(T1 )  pend(T2). Proof. T has A = fv : v 2 V (T ) deg(v)  3g =6 , and according to Lemma 2, W = T A] is a subtree of T . Let w1 w2 2 A be adjacent and such that w1 2 pend(W ). Then e = w1w2 2 E (T ), and e satises all the claims. Theorem 3. A tree T having at least two vertices is well-covered if and only if either T is a well-covered spider, or T is the internal edge-join of a number of well-covered spiders. Proof. We use induction on m = jfv : v 2 V (T ) deg(v)  3gj. If m = 0, then 1  deg(v)  2, for any v 2 V (T ), and T is isomorphic to either K2 or P4. If m = 1, then T is a well-covered spider, as well. Suppose the assertion is true for any tree having at most m ; 1 vertices of degree  3, and let T be a tree with m vertices of degree at least 3. According to Lemma 2, the subgraph W of T spanned by fv : v 2 V (T ) deg(v)  3g is a subtree of T . Let w1 w2 be two adjacent vertices of W such that w1 2 pend(W ). Hence, e = w1 w2 2 E (T ) T ; e = T1  T2 and T = T1 T2. Assume that wi 2 V (Ti ), for i = 1 2. Then, T1 is a well-covered spider, while T2 is a well-covered tree with at most m ; 1 vertices of degree  3. By induction hypothesis, T2 is either a well-covered spider or the internal edge-join of a number of well-covered spiders. Consequently, T has the same property. The converse is true by Proposition 2. As an example, the well-covered tree T in Figure 3 can be written as T = T P4 , and T is a well-covered spider with a vertex of degree 4. Sankaranarayana and Stewart have proposed in 17] another decomposition for a well-covered graph G, namely into a well-covered k-partite graph and the graph H = GN (S1 \ S2)] , where S1 S2 2 (G). By restricting H to be also well-covered, Sankaranarayana and Stewart have obtained two recursively decomposable subclasses of well-covered graphs. 0 0 6 Well-Covered Trees v v v v v v v v v v v v Figure 3: One reducible well-covered tree. 3. Distances in a well-covered tree Notice that the tree in Figure 4 has D(pend(T )) = f2 4g, while for the well-covered tree in Figure 3, D(pend(T )) = 3 6]. We show in the sequel that this fact is true for any well-covered tree. v v v v v v v Figure 4: D(v) = f2 4g, for any v 2 pend(T ). Theorem 4. If T is a well-covered tree with at least three vertices, then there exists 3  k 2 N , such that D(pend(T )) = 3 k]. Proof. We prove by induction on n = jV (T )j that there exists some 3  k 2 N , such that D(pend(T )) = 3 k]. Since a well-covered tree has a perfect matching, it follows that n must be even. The assertion is true for n = 4, since in this case k = 3. Suppose the statement is valid for any tree of even order less than n = 2m, and let T = (V E ) be a wellcovered tree with jV j = n > 4. According to Proposition 4, T can be written as T = T K2 , where K2 = (fa bg fabg), a 2 pend(T ) and T is a well-covered tree. By induction hypothesis, there exists 3  k 2 N , such that D(pend(T )) = 3 k]. Clearly, D(pend(T )) D(pend(T )), because pend(T ) pend(T ). We show that if D(pend(T )) ( D(pend(T )), then necessarily D(pend(T )) = 3 k + 1]. Suppose, on the contrary, that there are u v 2 pend(T ), such that d(u v)  k + 2. Since D(pend(T )) = 3 k], it follows that a 2 fu vg say u = a. Let xy 2 E be such that x 2 pend(T ) and yb 2 E . Since the unique path in T , connecting the vertices v and a, must contain y, we get d(v x)  k + 1, in contradiction with the assumption that D(pend(T )) = 3 k]. Consequently, we infer that either D(pend(T )) = 3 k] or D(pend(T )) = 3 k + 1], and this completes the proof. Remark 1. The converse of the Theorem 4 is not generally true. See Figure 5 for a tree T with D(pend(T )) = 3 5], and neither is well-covered, nor has a perfect matching. 0 0 0 0 0 0 0 0 7 Well-Covered Trees v @v@ v v @v v v v v Figure 5: A non-well-covered tree without perfect matching. Corollary 1. If T is a well-covered tree with at least three vertices, then D(pend(T )) = 3 diam(T )]: Proof. The result follows from Theorem 4, because for any tree there exist two vertices u v 2 pend(T ), such that d(u v) = diam(T ). Corollary 2. If T is a well-covered tree with at least three vertices, then there is some v 2 pend(T ) such that D(v) = 3 diam(T )]. Proof. Let x y 2 pend(T ) be such that d(x y) = diam(T ). Then, by choosing v 2 fx yg, we obtain the result. Remark 2. The tree T in Figure 6 has D(pend(T )) = 3 7], possesses a perfect matching, but is still non-well-covered. v v @@ v v @v v v v v v ;v ; v; v v Figure 6: A non-well-covered tree with a perfect matching. Remark 3. The tree T in Figure 7 has a perfect matching, is not wellcovered, but D(v) is an interval for any v 2 pend(T ), except that for some pendant vertices D(v) = 3 7], while for others D(v) = 4 6]. v @v@ v v @v v v v v v ;v ; ;v v v v v v v Figure 7: For every v 2 pend (T ), we see that D (v) = k1 (v) k2 (v)], but k1 (v) 2 f3 4g. Notice that in the case of the well-covered tree in Figure 3, for any v 2 pend(T ) there exists 3  k(v) 2 N , such that D(v) = 3 k(v)]. We show that this is not a symptomatic fact. 8 Well-Covered Trees Theorem 5. If is a well-covered tree with at least three vertices, then for every 2 ( ), there exists 3  ( ) 2 N , such that ( ) = 3 ( )]. Proof. If 2 ( ) and 2 ( ) ; f g, then ( )  3, since j ( )j  3, and for ( ) = 2, we get that ( ) \ ( ) = f g, and T u pend T k u u V pend T T v D u pend T u d u v N v k u d u v N u w consequently no stability system of T contains w, contradicting the fact that T is well-covered. Let  = (T ). According to Theorem 2, we get that jpend(T )j = . If pend(T ) = fui : 1  i  g and N (pend(T )) = fwi : 1  i  g, it follows that M = fuiwi : 1  i  g is the unique perfect matching of T . Assume that u = u1, d(u1 u2) = minfd(u1 ui) : 2  i  g, and d(u1 u ) = maxfd(u1 ui ) : 1  i  g. Then d(u1 u2 ) = 3, since otherwise the path connecting u1 and u2 contains w2 w3 and at least another wi . Hence, d(u1 ui ) < d(u1 u2 ), in contradiction with the choice of u2 . Suppose that for any i 2 f3 ::: j g, there is some v 2 pend(T ) such that d(u1 v) = i, but for j + 1 < d(u1 u ) there is no v 2 pend(T ) with d(u1 v ) = j + 1. Let r be the rst integer greater than j + 1, such that there exists ui 2 pend(T ) with d(u1 ui) = r  d(u1 u ). The unique path P from u1 to ui consists of only non-pendant vertices, except u1 and ui. Clearly, P contains wi too. Let wq be the second neighbor of wi on P . Hence, if uq 2 pend(T ) then j + 1  r ; 1 = d(u1 uq ) < r. By assumption on j + 1, the equalities j + 1 = r ; 1 = d(u1 uq ) are not valid. It implies that j + 1 < r ; 1 = d(u1 uq ) < r, which contradicts the conditions of our choice of r. Therefore, there exists 3  k 2 N , such that fd(u v) : v 2 pend(T ) u 6= vg = 3 k]. Corollary 3. If is a well-covered tree with at least three vertices and 2 ( ), then ( ) = 3 ], where = maxf ( ) : 2 ( )g. Remark 4. The converse of Theorem 5 is not generally true. See the trees T u pend T D u k k d u v v pend T in Figure 8 and Figure 9. v v v v v v v v v v v v v v v v v v v v Figure 8: D(v) = 3 k (v)], for any matching. v 2 v v v v v v ( ), and pend T T has a perfect 9 Well-Covered Trees v v @@ v v @v v v v ;v ; ;v v v Figure 9: D(v) = 3 k (v)], for any matching. v 2 v ( ), and pend T T has no perfect In the following proposition we completely describe well-covered spiders. Proposition 6. A tree spider if and only if ( T having at least two vertices is a well-covered ( )) 2 f1 1] 3 3] 3 4]g. D pend T Proof. Let T be a well-covered spider. If T has no vertices of degree greater than 2, then either T is isomorphic to K2 and D(pend(T )) = 1 1], or T is isomorphic to P4 and D(pend(T )) = 3 3]. Suppose that T has a vertex u with deg(u)  3. If N (u) \ pend(T ) = fwg, then d(w v) = 3 for any v 2 pend(T ) ; fwg, while d(v1 v2) = 4 for any v1 v2 2 pend(T ) ; fwg. Hence, it follows that D(pend(T )) = 3 4]. Conversely, if D(pend(T )) = 1 1], then T is isomorphic to K2 . Let D (pend(T )) = 3 3]. Then there exist u v 2 pend(T ) with d(u v ) = 3. For any other pendant vertex w, if such a vertex exists in T , we obtain either d(u w) = 3 and d(v w) 2 f2 4g, or d(v w) = 3 and d(u w) 2 f2 4g, which contradicts the assumption D(pend(T )) = 3 3]. Consequently, T is isomorphic to P4. Suppose that D(pend(T )) = 3 4], but T is not a well-covered spider. Clearly, T is neither K2 , nor P4. Assume that T is not a spider, i.e., there are at least two vertices w1 w1 of degree  3, joined by a path P containing at least an edge. Let v1 v2 2 pend(T ) be such that N (vi ) \ N (wi ) = fuig i = 1 2, and none of u1 u2 is on P , which must exist, since deg(wi )  3 i = 1 2. Hence we obtain that d(v1 v2)  5, in contradiction with the assumption on T . Therefore, T is a spider. Now we have to show that T is well-covered. Let u 2 V (T ) be with deg (u)  3. Then, any path joining pendant vertices in T must contain u. Since there exist v1 v2 2 pend(T ) with d(v1 v2 ) = 3, we may suppose that v1 2 N (u). N (u) \ pend(T ) = fv1g, because for any other vertex v 2 N (u) \ pend(T ) ; fv1 g, we get d(v1 v ) = 2, which contradicts the premise on T . Since deg(u)  3, it follows that pend(T );fv1 v2g 6= . If v 2 pend(T ); fv1 v2 g then d(v1 v) = 3, because otherwise d(v2 v)  5, in contradiction with the premise on T . Consequently, we get also that d(v2 v) = 4. In conclusion, for any v 2 pend(T );fv1 g, we get d(v1 u) = 1 and d(v u) = 2, i.e., T is well-covered. 10 Well-Covered Trees Proposition 7. If is a well-covered tree with j ( )j  3, then for every 2 ( ); ( ) there is 1 ( ) 2 N , such that ( ) = 1 ( )]. Proof. Let 2 ( ) be the neighbor of . By Theorem 5, there is some 3  1 2 N , such that f ( ) : 2 ( ) 6= g = 3 1]. Consequently, we get that f ( ) : 2 ( )g = 1 1 ; 1] = 1 ], and, clearly, = 1 ; 1 1. Remark 5. The converse of the above proposition is not generally true. T u V T V pend T w < k k k u u d w v d u v k D u pend T k T u v v pend T pend T v w k k k > See, for instance, the graph in Figure 10. v v v v v v v v v Figure 10: For every u 2 V (T ) ; pend (T ) D ( ) = 1 ( )]. u k u Theorem 6. If T is a tree with jV (T )j  3 and for every v 2 V (T ) there exists k(v) 2 N , such that D(v) = 3 k(v)], whenever v 2 pend(T ), and D (v ) = 1 k (v )], provided v 2 V (T ) ; pend(T ), then T is a well-covered graph. Proof. According to Theorem 2, it suces to show that T has a perfect matching consisting of only pendant edges. If v 2 V (T ) ; pend(T ), then there is some w 2 pend(T ), with d(v w) = 1. On the other hand, d(w u)  3 holds for any u 2 pend(T ) ; fw g. Consequently, the set f(v w) : v 2 V (T ) ; pend(T ) w 2 pend(T ) d(v w) = 1g is a perfect matching T consisting of pendant edges. Corollary 4. If T is a tree with jV (T )j  3 and for every v 2 V (T ) there exists k(v) 2 N , such that D(v) = 3 k(v)], whenever v 2 pend(T ), and T is simplicial, then T is well-covered. Proof. Since T is simplicial, jN (u) \ pend(T )j  1 holds for any u 2 V (T ) ; pend(T ), i.e., for every nonpendant vertex u, D (u) = 1 k (u)], where 2  k(u) 2 N . Together with Theorem 6 it brings us to the conclusion that T is a well-covered tree. Combining Theorem 5, Theorem 6 and Corollary 4 we get: Theorem 7. If T is a tree with at least three vertices, then the following conditions are equivalent: (i ) T is well-covered (ii ) for each v 2 V (T ) there is k(v) 2 N , such that D(v) = 3 k(v)], whenever v 2 pend(T ), and T is simplicial (iii ) for each v 2 V (T ) there is k(v) 2 N , such that D(v) = 3 k(v)], whenever v 2 pend(T ), and D(v) = 1 k(v)], provided v 2 V (T ); pend(T ). Well-Covered Trees 4. 11 Conclusions In this paper we study distances between the vertices and the pendant vertices of a well-covered tree. We see that distances between pendant vertices of a tree cannot completely describe its structure as a well-covered tree. It seems to be interesting to characterize these non-well-covered trees. It also seems that a number of results we have obtained on well-covered trees can be extended to well-covered graphs of girth greater than 5 and non-isomorphic to C7 and K1 . References 1] C. Berge, Graphs, North-Holland, Amsterdam, 1985. 2] S. R. Campbell, M. N. Ellingham and G. F. 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