Differential of a graph and its relation to the concept of domination

Abstract

Let G = (V (G),E(G)) be an arbitrary graph. For X ⊆ V (G), the boundary B(X) of X is the set of vertices in V (G)\X that have a neighbor in X and the differential of X is ∂(X) = |B(X)| − |X|. The differential ∂(G) of graph G is ∂(G) = max{∂(X) : X is a subset of V (G)}. It is shown in this paper that for any graph G of order n ≥ 2, ∂(G) is between 0 and (n − 2). Graphs whose differential is |V (G)| − 2 are characterized. Differentials of the join, corona, and lexicographic product of two graphs are also determined. For the lexicographic product G[H] of two graphs, it is shown that if the order of H is |V (H)| − 2, then one half of the difference of the order of G[H] and its differential is equal to its domination number. Otherwise, this difference is the total domination of G[H]. Mathematics Subject Classification: 05C69