• THE FORCING MONOPHONIC NUMBER OF A GRAPH

Abstract

For a connected graph G = (V,E), let a set S be a minimum monophonic set of G. A subset T  S is called a forcing subset for S if S is the unique minimum monophonic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing monophonic number of S, denoted by fm(S), is the cardinality of a minimum forcing subset of S. The forcing monophonic number of G, denoted by fm(G), is fm(G)=min{fm(S)}, where the minimum is taken over all minimum monophonic sets in G. Some general properties satisfied by this concept are studied. The forcing monophonic number of certain classes of graphs are determined. It is shown that for every integers a and b with 0≤aa+1, there exists a connected graph G such that fm(G)=a and m(G) = b.