# 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 \( \lvert V(G) \rvert = r\).

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