Forcing Total Detour Monophonic Sets in a Graph

Abstract

For a connected graph G = (V,E) of order at least two, a total
detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A subset T of a minimum total detour monophonic set S of G is a forcing total detour monophonic subset for S if S is
the unique minimum total detour monophonic set containing T. A forcing total detour monophonic subset for S of minimum cardinality is a minimum forcing total detour monophonic subset of S. The forcing total detour monophonic number ftdm(S) in G is the cardinality of a minimum forcing total detour monophonic subset of S. The forcing total detour monophonic number of G
is ftdm(G) = min{ftdm(S)}, where the minimum is taken over all minimum total detour monophonic sets S in G. We determine bounds for it and find the forcing total detour monophonic number of certain classes of graphs. It is
shown that for every pair a, b of positive integers with 0 6 a < b and b > 2a+1, there exists a connected graph G such that ftdm(G) = a and dmt(G) = b.