The forcing connected vertex monophonic number of a graph

Abstract

For any vertex x in a connected graph G of order p ≥ 2, a set Sx ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x - y monophonic path for some element y in Sx. The minimum cardinality of an x-monophonic set of g is the x-monophonic set of G is an x-monophonic set Sx such that the subgraph G［Sx］induced by Sx is connected. The minimum cardinality of a connected x-monophonic set of G is defined as the connected x-monophonic number of G and is denoted by cmx(G). A subset T of a minimum connected x-monophonic set Sx of G is called an x-forcing subset for Sx if Sx is the unique minimum connected x-monophonic set containing T. An x-forcing subset for Sx of minimum cardinality is a minimum x-forcing subset of Sx. The forcing connected x-monophonic number of Sx, denoted by fcmx(Sx), is the cardinality of a minimum x-forcing subset for Sx. The forcing connected x-monophonic number of G is fcmx (G) = min ｛fcmx (Sx)｝, where the minimum is taken over all minimum connected x-monophonic sets Sx in G. We determine bounds for it and find the forcing connected vertex monophonic number for some special classes of graphs. For any two positive integers a and b with 0≤ a < b-1, there exists a connected graph G with fcmx(G) = a and cmx (G) =b for some vertex x in G.