On the D-differential of a graph

Abstract

AbstractLet [Formula: see text] be a graph of order n(G). For a subset S of V(G), the boundary of S is defined as [Formula: see text] where N(S) is the open neighborhood of S. The external private neighborhood set of v with respect to S is defined as [Formula: see text] For a subset S of V(G), the D-differential of S is defined as [Formula: see text] where [Formula: see text] In this paper, we introduce the concept of D-differential of a graph G, which is defined as [Formula: see text] We present several lower and upper bounds of D-differential of a graph. We construct a Gallai-type theorem for the D-differential [Formula: see text] and double Roman domination number [Formula: see text] which states that [Formula: see text] Thus, we can utilize a relation between D-differential and double Roman domination number. The concept of D-differential can be a framework to find double Roman domination number of graphs. Actually, we determine the double Roman domination number of middle graphs.

Keywords

• Differential
• double Roman domination
• middle graph
• D-differential
• 05C69