On the Forcing Hull and Forcing Monophonic Hull Numbers of Graphs

Abstract

For a connected graph G = (V,E), let a set M be a minimum monophonic hull set of G. A subset T ⊆ M is called a forcing subset for M if M is the unique minimum monophonic hull set containing T. A forcing subset for M of minimum cardinality is a minimum forcing subset of M. The forcing monophonic hull number of M , denoted by fmh(M), is the cardinality of a minimum forcing subset of M. The forcing monophonic hull number of G, denoted by fmh(G), is fmh(G) = min {fmh(M)}, where the minimum is taken over all minimum monophonic hull sets in G. Some general properties satisfied by this concept are studied. Every monophonic set of G is also a monophonic hull set of G and so mh(G) ≤ h(G), where h(G) and mh(G) are hull number and monophonic hull number of a connected graph G. However, there is no relationship between fh(G) and fmh(G), where fh(G) is the forcing hull number of a connected graph G. We give a series of realization results for various possibilities of these four parameters.