Excellent graphs Preethi Kuttipulackal Mixed Tree Domination in Graphs Thesis. Department of Mathematics, University of Calicut, 2012

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1 Excellent graphs Preethi Kuttipulackal Mixed Tree Domination in Graphs Thesis. Department of Mathematics, University of Calicut, 2012

2 CHAPTER 5 Excellent graphs 5.1 Introduction The main difficulty in the direct search of a γ mt -set in a graph is the uncertainty that where to start with. For example, if G has a pendant vertex v, then the support vertex adjacent to v or the only edge incident at v should be in any γ mt -set, so that we can start with one of them. Even we get an easy start, the succeeding steps may not be easy. If a graph G is such that, any vertex or edge of it is in some γ mt -set, then we can start at any element. This chapter deals with such graphs. 80

3 5.2 Definition And Properties It was Claude Berge in 1980[9] who introduced the concept of B-graphs, that is graphs in which every vertex is contained in a maximum independent set. Frick et. al. in 2002 made a beginning of the study of graphs which are excellent with respect to various graph parameters. A graph G is γ-excellent if each of its vertex belongs to some γ-set of G. Continuing the study on γ-excellent graphs, N. Sridharan and Yamuna made an extensive work in this area. We generalize this concept in view of mixed tree domination. Definition A graph G is said to be mt-excellent if every element in V E belongs to some γ mt -set of G. mt-excellent graphs Graphs that are not mt-excellent Figure 5.1: 81

4 Definition A graph G is said to be mt-vertex excellent if every vertex in G belongs to some γ mt -set of G. Definition A graph G is said to be mt-edge excellent if every edge in G belongs to some γ mt -set of G. It is obvious that every mt- excellent graph is mt-vertex excellent and mtedge excellent. Because of the symmetry in adjacency and incidence, complete graphs, cycles, and n-cubes are all mt-excellent. For any n > 2, the path P n is not mt excellent, because its end vertices will not be in any γ mt -set of P n. But for n 3, P n is mt-edge excellent. In fact, any graph, on more than two vertices, having an end vertex cannot be mt-excellent. We prove a stronger result. Theorem If G is an mt-vertex excellent graph on n 3 vertices, then every vertex in G is contained in a cycle. Proof. Let G be mt- vertex excellent. Then G cannot have a pendant vertex, because no γ mt -set of G will contain a pendant vertex. If possible let v, be a vertex in G which is not a part of any cycle in G. Then none of the edges incident at v is a part of a cycle, that is, they are all cut edges. So v is a cut vertex. Since G has no end vertex, G has no pendant edge. Therefore by Theorem 2.2.5, every γ mt -set of G must contain all edges incident at v. But then v is not required in any γ mt -set of G so that G cannot be mt-excellent, a contradiction. Hence every vertex is contained in a cycle. 82

5 As a consequence of the theorem we have the following results: (i) An mt-vertex excellent graph is 2-connected. (ii) In an mt-vertex excellent graph, if two blocks B 1 and B 2 are connected by a unique nontrivial (u, v)-path P, where V (P ) V (B 1 ) = {u} and V (P ) V (B 2 ) = {v}, then the path P must be P 2 = uv. Since mt-excellent graphs are mt-vertex excellent, all the results about mtvertex excellent graphs also hold in mt-excellent graphs. For the last graph in Figure 5.1, each edge is contained in a cycle but the graph has no γ mt -set containing the vertices of degree 2, which shows that the 2-connectivity of graph is not a sufficient condition for mt-excellence. But, if G is a block with γ mt (G) = n 1, this becomes a sufficient condition for mt-excellence. We need the following two results. Lemma For any edge e of a connected graph G, there is a spanning tree of G containing e. Proof. If T is a spanning tree of G, which does not contain the edge e of G, then T + e will contain a unique cycle and e will be a part of this cycle. If e is an edge of this cycle other than e, then (T + e) e is a spanning tree of G containing e. Now Lemma proves: Theorem Any graph G with γ mt (G) = n 1, is mt-edge excellent. Lemma If G is a block, then for every edge e, G has a spanning tree having e as a pendant edge. 83

6 Proof. Given an edge e of G, by the Lemma 5.2.1, G has a spanning tree T containing e. If e is not a pendant edge of T, consider an edge f adjacent to e in T. Since G is a block, there is a cycle C in G containing both e and f. Form the graph G 1 by adding the edges of E(C) \ E(T ) to T. Removing f and one edge other than e from each cycle of G 1, we arrive at a spanning tree T 1 of G. e e f v e v E(C)-E(T) T G 1 T 1 Figure 5.2: If v is the vertex common to e and f, the addition of edges of E(C) \ E(T ) in T will not change the degree of v in T. That is deg G1 (v) = deg T (v). Also, because of the construction of T 1, deg T1 (v) = deg T (v) 1. Now if deg T1 (v) = 1, then e is a pendant edge in T 1 as required. Otherwise as 84

7 T 1 constructed from T, we can construct a spanning tree T 2 where deg T2 (v) = deg T1 (v) 1 by keeping the edge e in T 2. Here the 2-connectivity is strongly used and the selection of the edge f ensures that an edge(adjacent to e) removed once, will never return in any successive steps. Because, at each step, we are considering a cycle containing two given edges having the common vertex v, so that this cycle will not contain any other edge at v. Since the graph is finite, the process will terminate in a finite number of steps, where we get a spanning tree T k of G containing e, with deg Tk (v) = 1. In the above proof, since the graph is a block, instead of reducing the degree at v, the degree at the other end of e can be reduced as well. Lemma If G is a block, then for any vertex v, G has a spanning tree having v as a pendant vertex. Now we are in a position to prove the theorem. Theorem A block G with γ mt (G) = n 1 is mt-excellent. Proof. Since γ mt (G) = n 1, Lemma shows that G is mt-edge excellent. So, it is enough to prove that: for any vertex v, G has a γ mt -set containing v. Let w be a vertex adjacent to v in G. By Lemma and 5.2.3, there exists a spanning tree T of G containing the edge vw as a pendant edge and w as a 85

8 pendant vertex. Then the edges in E(T )\{vw} together with the vertex v form a γ mt -set. The first graph in Figure 5.1 is not a block, but it is mt-excellent and γ mt (G) = n 1. The path P n, n > 3(which is not a block) is an example for a graph with γ mt (P n ) = n 1, but not mt-excellent. The graph K 4 e(the last graph in Figure 5.1) is an example for a block with γ mt (K 4 e) < n 1, which is not mt-excellent. We hope that regular blocks are mt-excellent. Figure 5.3 shows two nonisomorphic 3-regular blocks of order 6, both are mt-excellent. Also, they provide examples for mt-excellent graphs with γ mt (G) < n 1. Figure 5.3: Figure 5.4 shows that there are regular graphs which are not mt-excellent. This graph has no γ mt -set containing the vertex v, because every edge adjacent to v, which are all cut edges but not pendant edges, should be in every γ mt -set. 86

9 v Figure 5.4: Theorem A tree T is mt-edge excellent if and only if the number of pendant vertices is equal to the number of support vertices. Proof. If the number of pendant vertices is equal to the number of support vertices in a tree T, by Theorem 3.2.4, γ mt (G) = n p + s 1 = n 1, and so G is mt-excellent. Conversely, if the number of pendant vertices is not equal to the number of support vertices in a tree T, then T has a support vertex v which is adjacent to at least two pendant vertices. Then v should be in every γ mt -set of T, so that the pendant vertices adjacent at v will not be a part of any γ mt -set of T. Example An mt-edge excellent graph need not be mt-vertex excellent. For example, the path P n, n > 3 is mt-edge excellent but not mt-excellent. Also, an mt-vertex excellent graph need not be mt-edge edge excellent. For example, the wheel graph on five vertices W 5 is mt-vertex excellent but not mt-edge excellent. 87

10 5.3 Open Problems Design an algorithm for finding a γ mt -set of an mt-excellent graph. Characterize mt-excellent(mt-edge as well as mt-vertex excellent) graphs Is every regular block mt-excellent? Can we get a nice formula of γ mt (G) for mt-excellent graphs? If G is mt-excellent, then is there any speciality for the automorphism group of G? 88