Forcing (G,D)-number of a Graph

Abstract

In (7), we introduced the new concept (G,D)-set of graphs. Let G = (V, E) be any graph. A (G,D)-set of a graph G is a subset S of vertices of G which is both a dominating and geodominating(or geodetic) set of G. The minimum cardinality of all (G,D)-sets of G is called the (G,D)-number of G and is denoted by γG(G). In this paper, we introduce a new parameter called forcing (G,D)-number of a graph G. Let S be a γG-set of G. A subset T of S is said to be a forcing subset for S if S is the unique γG-set of G containing T. A forcing subset T of S of minimum cardinality is called a minimum forcing subset of S. The forcing (G,D)-number of S denoted by fG,D(S) is the cardinality of a minimum forcing subset of S. The forcing (G,D)-number of G is the minimum of fG,D(S), where the minimum is taken over all γG-sets S of G and it is denoted by fG,D(S).