%0 Journal Article
%T Extreme outer connected monophonic graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A K., Ganesamoorthy
%A S, Lakshmi Priya
%D 2022
%\ 12/01/2022
%V 7
%N 2
%P 211-226
%! Extreme outer connected monophonic graphs
%K outer connected monophonic set
%K outer connected monophonic number
%K extreme order
%K extreme outer connected monophonic graph
%R 10.22049/cco.2021.27042.1184
%X 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 $k$.
%U http://comb-opt.azaruniv.ac.ir/article_14268_9a46437c0806e08d6d1d2388dd680b82.pdf