Minimum Diameter Spanning Tree

In this paper, we discuss the simple connected graphs which have a minimum diameter spanning tree such that both have same domination number.


Preliminaries
E.J. Cockayne and S.T. Hedetniemi [1] introduced the concept dominating set. Frank Harary, Robert Z.Norman and Dorwin Cartwright [2] explained an interesting application in voting situations using the concept of domination. C.L.
Liu [3] also discussed the application of dominance to communication network, where a dominating set represents a set of cities which acting as transmitting stations, can transmit messages to every city in the network.
A subset S of vertices from V is called a dominating set for G if every vertex of G is either a member of S or adjacent to a member of S. A dominating set of G is called a minimum dominating set if G has no dominating set of smaller cardinality. The cardinality of minimum dominating set of G is called the dominating number for G and it is denoted by ( ) G γ [4].
The eccentricity of a graph vertex v in a connected graph G is the maximum graph distance between v and any other vertex u of G. The eccentricity ( ) . The radius of a graph ( ) M. Yamuna and K. Karthika [5] provided a constructive procedure to generate a spanning tree for any graph from its dominating set, γ-set.
In this paper, we discuss few simple connected graphs for which the domination numbers of the graph and that of its minimum diameter spanning tree are the same.

Some Definitions
In this section, we present few definitions and examples which are required for this article. Definition 1. The spanning tree T of the simple connected graph G is said to be a minimum diameter spanning tree if there is no other spanning tree T' of G A vertex u of a simple connected graph is an essential dominating vertex, if every minimum dominating set contains the vertex u. Example 1. In the graph given in Figure 1, the vertex u is an essential dominating vertex. Definition 3. [6] The pan graph is the graph obtained by joining a cycle graph C n to a singleton graph K 1 with a bridge and we denote it by n  . Example 2. The graphs given in Figure 2 are pan graphs. Definition 4. [6] The ladder graph L n is a planar undirected graph with 2n vertices and ( ) Example 3. The graph given in Figure 3 is a ladar graph L 7 .   V. T. Chandrasekaran, N. Rajasri

Some Results
In this section we discuss the graphs which have minimum diameter spanning trees such that both have same domination number. It is clear to observe that cycle, regular graph, complete bipartite graph have minimum diameter spanning tree, possessing domination number same that of the graph. Proposition 1. Every pan graph has at least one essential dominating vertex. In fact, the vertex, which is adjacent to the pendent vertex is an essential dominating vertex.
Proof: By definition 3 of a pan graph, it is a cycle in which one of the nodes in the cycle is adjacent to a node which is not in the cycle. Let u be the such vertex in the cycle, which is adjacent to the pendent vertex v. Since u is in the cycle, u is also adjacent to two other vertices, say x and y, in the cycle. Clearly, the vertex u dominates the three vertices v, x and y. Suppose u is not an essential dominating set, then there must exist another vertex in the cycle, which dominates the four vertices u, v, x and y. But obviously, no such vertex exists. Hence, u is an essential dominating vertex.
Let us recall the following results ( ) Hence the proof of the proposition follows.
Proposition 3. The ladder graphs L n have minimum diameter spanning tree T such that ( ) ( ) n L T γ γ = , only when 3 n ≤ or n is an even integer. Proof: In a ladar graph L n , the 2n vertices are divided in to two sets of each n vertices and each vertex of one set is adjacent to exactly one vertex of the other set and vice-versa. We may call these edges as step edges. Further, each of the n vertices are connected by a path on 1 n − edges. We may note that the diameter of a minimum diameter spanning tree of L n is n, when n is odd and 1 n + , when n is even. In fact when n is odd, L n has a unique minimum diameter spanning tree, which is obtained by deleting all the step edges, except the middle one.
When n is even, there may be several minimum diameter spanning trees. The V. T. Chandrasekaran, N. Rajasri