Cubo (Dec 2020)
Characterization of Upper Detour Monophonic Domination Number
Abstract
This paper introduces the concept of \textit{upper detour monophonic domination number} of a graph. For a connected graph $G$ with vertex set $V(G)$, a set $M\subseteq V(G)$ is called minimal detour monophonic dominating set, if no proper subset of $M$ is a detour monophonic dominating set. The maximum cardinality among all minimal monophonic dominating sets is called \textit{upper detour monophonic domination number} and is denoted by $\gamma_{dm}^+(G)$. For any two positive integers $p$ and $q$ with $2 \leq p \leq q$ there is a connected graph $G$ with $\gamma_m (G) = \gamma_{dm}(G) = p$ and $\gamma_{dm}^+(G)=q$. For any three positive integers $p, q, r$ with $2 < p < q < r$, there is a connected graph $G$ with $ m(G) = p $, $\gamma_{dm}(G) = q $and$ \gamma_{dm}^+(G)= r $. Let $ p $ and $ q $ be two positive integers with $ 2 < p<q $ such that $ \gamma_{dm}(G) = p $ and $ \gamma_{dm}^+(G)= q $. Then there is a minimal DMD set whose cardinality lies between $ p $ and $ q $. Let $ p , q $ and $ r $ be any three positive integers with $ 2 \leq p \leq q \leq r$. Then, there exist a connected graph $ G $ such that $ \gamma_{dm}(G) = p , \gamma_{dm}^+(G)= q $ and $ \lvert V(G) \rvert = r$.
Keywords
