THE FORCING CONVEX DOMINATION NUMBER OF A GRAPH

Abstract

Let G be a connected graph and D a minimum convex domination set of G. A subset T ⊆ D is called a forcing subset of D, if D is the unique minimum convex dominating set containing T. A forcing subset for D of minimum cardinality is a minimum forcing subset of D. The forcing convex domination number of D, denoted by γcon(D), is the cardinality of a minimum forcing subset of D. The forcing convex domination number of G, denoted by fγcon(G) and is defined by fγcon(G) = min {fγcon(D)}, where the minimum is taken over all minimum convex dominating sets D in G. Some general properties satisfied by this concepts are studied. The forcing fair dominating number of certain standard graphs are determined. It is shown that for every pair a, b of integers with 0 ≤ a <b, there exists a connected graph G such that fγcon(G) = a and γcon(G) = b.