Closed Monophonic and Minimal Closed Monophonic Numbers of a Graph

Abstract

This paper considered the concepts of monophonic, closed monophonic, and minimal closed monophonic numbers of a connected graph $G$. It was shown that any positive integers $m,n,d$, and $k$ satisfying the conditions that $4\le n\le m,3\le d\le k$, and $k\ge 2m–n+d+1$ are realizable as the monophonic number, closed monophonic number, $m$-diameter, and order, respectively, of a connected graph. Also, any positive integers $n,m,d$, and $k$ with $2\le n\le m,d\ge 3$, and $k\ge m+d–1$ are realizable as the closed monophonic number, minimal closed monophonic number, $m$-diameter, and order, respectively, of a connected graph. Further, the closed monophonic number of the composition of connected graphs was also determined.