Forcing total outer connected monophonic number of a graph

Abstract

For a connected graph $G = (V,E)$ of order at least two, a subset $T$ of a minimum total outer connected monophonic set $S$ of $G$ is a forcing total outer connected monophonic subset for $S$ if $S$ is the unique minimum total outer connected monophonic set containing $T$. A forcing total outer connected monophonic subset for $S$ of minimum cardinality is a minimum forcing total outer connected monophonic subset of $S$. The forcing total outer connected monophonic number $f_{tom}(S)$ in $G$ is the cardinality of a minimum forcing total outer connected monophonic subset of $S$. The forcing total outer connected monophonic number of $G$ is $f_{tom}(G) =  \min\{f_{tom}(S)\}$, where the minimum is taken over all minimum total outer connected monophonic sets $S$ in $G$.  We determine bounds for it and find the forcing total outer connected monophonic number of a certain class of graphs.  It is shown that for every pair $a,b$ of positive integers with $0 \leq a < b$ and $b \geq a+4$, there exists a connected graph $G$ such that $f_{tom}(G) = a$ and $cm_{to}(G) = b$, where $cm_{to}(G)$ is the total outer connected monophonic number of a graph.

Keywords

• total outer connected monophonic set
• total outer connected monophonic number
• forcing total outer connected monophonic subset
• forcing total outer connected monophonic number