Locating Equitable Domination and Independence Subdivision Numbers of Graphs

Abstract

Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wV- D, N(u)D  N(w)D, N(u)D=N(w)D. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of edges that must be subdivided(where each edge in G can be subdivided at most once) in order to increase the locating equitable domination number and is denoted by sdle(G). The independence subdivision number sdle(G) to equal the minimum number of edges that must be subdivided in order to increase the independence number. In this paper, we establish bounds on sdle(G) and sdle(G) for some families of graphs. For notation and graph theory terminology, we in general follow (3). Specifically, a graph G is a finite nonempty set V(G) of objects called vertices together with a possibly empty set E(G) of 2- element subsets of V(G) called edges. The order of G is n(G) = V(G) and the size of G is m(G) = E(G) . The degree of a vertex vV(G) in G is dG(v) = NG(v) . A vertex of degree one is called an end-vertex. The minimum and maximum degree among the vertices of G is denoted by (G) and (G), respectively. Further for a subset S  V(G), the degree of v in S, denoted ds(v), is the number of vertices in S adjacent to v; that is,ds(v) =N(v)S . In particular, dG(v) = dv(v). if the graph G is clear from the context, we simply write V,E,n,m,d(v),  and  rather than V(G),E(G),n(G),m(G),dG(v), (G) and (G), respectively. The closed neighborhood of a vertex uV is the set N(u)= {u}{v/uv}. Given a set S  V of vertices and a vertex uS, the private neighbor set of u, with respect to S, is the set pn(n,S) = N(u) - N( S - {u}). We say that every vertex vpn(u,S) is a private neighbor of u with respect to S. Such a vertex v is adjacent to u but is not adjacent to any other vertex of S, then it is an isolated vertex in the subgraph G(S) induced by S. In this case, upn(u,S), and we say that u is its own private neighbor. We note that if a set s is a (G)-set, then for every vertex uS, pn(u,S)  , i.e., every vertex of S has at least one private neighbor. It can be seen that if S is a (G)-set, and two vertices u,v S are adjacent, then each of u and v must have a private neighbor other than itself. We will slao use the following terminology. Let vV be a vertex of degree one; v is called a leaf. The only vertex adjacent to a leaf, say u, is called a support vertex, and the edge uv is called a pendant edge.Two edges in a graph G are independent if they are not adjacent in G. The distance dG(u,v) or d(u,v) between two vertices u and v in a graph G, is the length of a shortest path connecting u and v. The diameter of a connected graph G is defined to be max {dG(u,v) :u,vV(G) }. A set DV of vertices is a dominating set if every vertex in V-D is adjacent to