On The Detour Monophonic Number of a Graph

Abstract

For a connected graph $G=(V,E)$ of order at least two, a chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A longest $x$-$y$ monophonic path is called an $x$-$y$ detour monophonic path. A set $S$ of vertices of $G$ is a detour monophonic set of $G$ if each vertex $v$ of $G$ lies on an $x$-$y$ detour monophonic path for some $x$ and $y$ in $S$. The minimum cardinality of a detour monophonic set of $G$ is the detour monophonic number of $G$ and is denoted by $dm(G)$. For any two vertices $u$ and $v$ in $G$, the monophonic distance $dm(u,v)$ from $u$ to $v$ is defined as the length of a $u$-$v$ detour monophonic path in $G$. The monophonic eccentricity $em(v)$ of a vertex $v$ in $G$ is the maximum monophonic distance from $v$ to a vertex of $G$. The monophonic radius $ra{d}_m(G)$ of $G$ is the minimum monophonic eccentricity among the vertices of $G$, while the monophonic diameter $dia{m}_m(G)$ of $G$ is the maximum monophonic eccentricity among the vertices of $G$. It is shown that for positive integers $r$, $d$, and $n\ge 4$ with $r<d$, there exists a connected graph $G$ with $ra{d}_m(G)=r$, $dia{m}_m(G)=d$, and $dm(G)=n$. Also, if $p$, $d$, and $n$ are integers with $2\le n\le p-d+4$ and $d\ge 3$, there is a connected graph $G$ of order $p$, monophonic diameter $d$, and detour monophonic number $n$. Further, we study how the detour monophonic number of a graph is affected by adding some pendant edges to the graph.