The nonsplit domination in subdivision graph

Abstract

A dominating set D of a graph G = (V, E) is a nonsplit dominating set if the induced subgraph ?V ? D? is connected. The nonsplit domination number ?ns(G) of G is the minimum cardinality of a nonsplit dominating set. An edge e = uv of a graph G is said to be subdivided if e is replaced by the edges uw and vw for some vertex w not in V (G). The graph obtained from G by subdividing each edge of G exactly once is called the subdivision graph of G and is denoted by S(G). In this paper, we study the nonsplit domination number of subdivision graph. We determine exact values of the nonsplit domination number of subdivision graph for some standard graphs. We also obtain bounds and relationship with other graph theoretic parameters for the ?ns(S(G)).