# Extreme outer connected monophonic graphs

Document Type : Original paper

Authors

1 Department of Mathematics, Coimbatore Institute of Technology (Government Aided Autonomous Institution) Coimbatore - 641 014, India

2 Department of Mathematics, Coimbatore Institute of Technology, Coimbatore

Abstract

For a connected graph $G$ of order at least two, a  set $S$ of vertices in a graph $G$ is said to be an \textit{outer connected monophonic set} if $S$ is a monophonic set of $G$ and either $S=V$ or the subgraph induced by $V-S$ is connected. The minimum cardinality of an outer connected monophonic set of $G$ is the \textit{outer connected monophonic number} of $G$ and is denoted by $m_{oc}(G)$.  The number of extreme vertices in $G$ is its \textit{extreme order} $ex(G)$.  A graph $G$ is said to be an \textit{extreme outer connected monophonic graph} if $m_{oc}(G)$ = $ex(G)$.  Extreme outer connected monophonic graphs of order $p$ with outer connected monophonic number $p$ and extreme outer connected monophonic graphs of order $p$ with outer connected monophonic number $p-1$ are characterized.  It is shown that for every pair $a, b$ of integers with $0 \leq a \leq b$ and $b \geq 2$, there exists a connected graph $G$ with $ex(G) = a$ and $m_{oc}(G) = b$. Also, it is shown that  for positive integers $r,d$ and $k \geq 2$ with $r < d$, there exists an extreme outer connected monophonic graph $G$ with monophonic radius $r$, monophonic diameter $d$ and outer connected monophonic number $k$.

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