The connected vertex detour monophonic number of a graph

Abstract

For any vertex x in a connected graph G of order $n \ge 2$, a set $S \subseteq V(G)$ is an x-detour monophonic set of G if each vertex $v \in V(G)$ lies on an x-y detour monophonic path for some element y in S. The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by $dm_x(G)$. A connected x-detour monophonic set of G is an x-detour monophonic set S such that the subgraph induced by S is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by $cdm_x(G)$. We determine bounds for it and find the value of $cdm_x(G)$ for some special classes of graphs. For positive integers r, d and k with $2 \le r \le d$ and $k \ge 2$, there exists a connected graph G with monophonic radius r, monophonic diameter d and connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j, k, l and n with $2 \le j \le k \le l \le n-4$, there is a connected graph G of order n with $m_x(G) = j, dm_x(G) = k$ and $cdm_x(G) = l$ for some vertex x in G, where $m_x(G)$ is the x-monophonic number of G.