On the upper open detour monophonic number of a graph

Abstract

An open detour monophonic set $M$ in a connected graph $G$ is called a minimal open detour monophonic set if no proper subset of $M$ is an open detour monophonic set of $G$. The upper open detour monophonic number $\operatorname{odm}^{+}(G)$ of $G$ is the maximum cardinality of a minimal open detour monophonic set of $G$. Some general properties satisfied by this concept are studied. The upper open detour monophonic number of some standard graphs are determined. Connected graphs of order $n$ with upper open detour monophonic number 2 or 3 or $n$ are characterized. It is shown that for every pair $a$ and $b$ of integers $a$ and $b$ with $2 \leq a \leq b$, there exists a connected graph $G$ such that $o d m(G)=a$ and $o d m^{+}(G)=b$, where $o d m(G)$ is the open detour monophonic number of a graph.