Minimal outer connected monophonic sets in graphs

Abstract

For a connected graph $G$ of order at least two, an outer connected monophonic set $S$ in a graph $G$ is called a minimal  outer connected monophonic set if no proper subset of $S$ is an outer connected monophonic set of $G$.  The upper outer connected monophonic number $m_{oc}^+(G)$ of $G$ is the maximum cardinality of a minimal outer connected monophonic set of $G$.  We determine bounds for it and find the upper  outer connected monophonic number of certain classes of graphs.  It is shown that for any three positive integers $a, b, c$ with $3 \leq a \leq b \leq c$, there is a connected graph $G$ with $m(G) = a, m_{oc}(G) = b, m_{oc}^+(G) = c$, where $m(G)$ is the monophonic number of a graph and $m_{oc}(G)$ is the outer connected monophonic number of a graph.  Also, it is shown that for any three positive integers $a, b$, and $n$ with $3 \leq a \leq n \leq b$, there is a connected graph $G$ with $m_{oc}(G) = a, m_{oc}^+(G) = b$, and a minimal outer connected monophonic set of cardinality $n$.