The forcing vertex detour monophonic number of a graph

Abstract

AbstractFor any two vertices x and y in a connected graph G, an x–y path is a monophonic path if it contains no chord, and a longest x–y monophonic path is called an x–y detour monophonic path. For any vertex x in G, a set Sx⊆V(G) is an x-detour monophonic set of G if each vertex v∈V(G) lies on an x–y detour monophonic path for some element y in Sx. The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dmx(G). A subset Tx of a minimum x-detour monophonic set Sx of G is an x-forcing subset for Sx if Sx is the unique minimum x-detour monophonic set containing Tx. An x-forcing subset for Sx of minimum cardinality is a minimum x-forcing subset of Sx. The forcing x-detour monophonic number of Sx, denoted by fdmx(Sx), is the cardinality of a minimum x-forcing subset for Sx. The forcing x-detour number of G is fdmx(G)=min{fdmx(Sx)}, where the minimum is taken over all minimum x-detour monophonic sets Sx in G. We determine bounds for it and find the same for some special classes of graphs. Also we show that for every pair s,t of integers with 2≤s≤t, there exists a connected graph G such that fdmx(G)=s and dmx(G)=t for some vertex x in G