Characterization of Upper Detour Monophonic Domination Number

Abstract

This paper introduces the concept of 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 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 \( \lvert V(G) \rvert = r\).