- Published 2015

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 number of G and is denoted by mx(G). A subset Tx of a minimum x-monophonic set Sx of G is an x-forcing subset for Sx if Sx is the unique minimum x-monophonic set containing Tx. An x-forcing subset for Sx of minimum cardinality is a minimum x-forcing subset of Sx. The forcing x-monophonic number of Sx, denoted by fmx(Sx), is the cardinality of a minimum x-forcing subset for Sx. The forcing x-monophonic number of G is fmx (G) = min{fmx (Sx)}, where the minimum is taken over all minimum x-monophonic sets Sx in G. We determine bounds for it and find the forcing vertex monophonic number for some special classes of graphs. It is shown that for any three positive integers a, b and c with 2 ≤ a ≤ b < c, there exists a connected graph G such that fmx (G) = a, mx(G) = b and cmx(G) = c for some vertex x in G, where cmx(G) is the connected x-monophonic number of G.

@inproceedings{Titus2015TheFV,
title={The Forcing Vertex Monophonic Number of a Graph},
author={P. Titus and Kuttalam Iyappan},
year={2015}
}