The forcing monophonic and forcing geodetic numbers of a graph

Abstract

For a connected graph G = (V, E), let a set S be a m-set of G. A subset T ⊆ S is called a forcing subset for S if S is the unique m-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. We know that m(G) ≤ g(G), where m(G) and g(G) are monophonic number and geodetic number of a connected graph G respectively. However there is no relationship between fm(G) and fg(G), where fg(G) is the forcing geodetic number of a connected graph G. We give a series of realization results for various possibilities of these four parameters.