The Upper Vertex Detour Monophonic Number of a Graph

Abstract

For any vertex \(x\) in a connected graph \(G\) of order \(n \geq 2\), a set \(S \subseteq V(G)\) is a \(z\)-detour monophonic set of \(G\) if each vertex \(v \in V(G)\) lies on a \(x-y\) detour monophonic path for some element \(y \in S\). The minimum cardinality of a \(x\)-detour monophonic set of \(G\) is the \(x\)-detour monophonic number of \(G\), denoted by \(dm_z(G)\). An \(x\)-detour monophonic set \(S_x\) of \(G\) is called a minimal \(x\)-detour monophonic set if no proper subset of \(S_x\) is an \(x\)-detour monophonic set. The upper \(x\)-detour monophonic number of \(G\), denoted by \(dm^+_x(G)\), is defined as the maximum cardinality of a minimal \(x\)-detour monophonic set of \(G\). We determine bounds for it and find the same for some special classes of graphs. For positive integers \(r, d,\) and \(k\) with \(2 \leq r \leq d\) and \(k \geq 2\), there exists a connected graph \(G\) with monophonic radius \(r\), monophonic diameter \(d\), and upper \(z\)-detour monophonic number \(k\) for some vertex \(x\) in \(G\). Also, it is shown that for positive integers \(j, k, l,\) and \(n\) with \(2 \leq j \leq k \leq l \leq n – 7\), there exists a connected graph \(G\) of order \(n\) with \(dm_x(G) = j\), \(dm^+_x(G) = l\), and a minimal \(x\)-detour monophonic set of cardinality \(k\).