Clique-to-vertex monophonic distance in graphs

Abstract

In this paper, we define the clique-to-vertex [Formula: see text]–[Formula: see text] monophonic path, the clique-to-vertex monophonic distance [Formula: see text], the clique-to-vertex monophonic eccentricity [Formula: see text], the clique-to-vertex monophonic radius [Formula: see text], and the clique-to-vertex monophonic diameter [Formula: see text], where [Formula: see text] is a clique and [Formula: see text] a vertex in a connected graph [Formula: see text]. We determine these parameters for some standard graphs. We show the inequality among the clique-to-vertex distance, the clique-to-vertex monophonic distance, and the clique-to-vertex detour distance in graphs. Also, it is shown that the clique-to-vertex geodesic, the clique-to-vertex monophonic, and the clique-to-vertex detour are distinct in [Formula: see text]. It is shown that [Formula: see text] for every connected graph [Formula: see text] and that every two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] are realizable as the clique-to-vertex monophonic radius and clique-to-vertex monophonic diameter of some connected graph. Also, it is shown any three positive integers [Formula: see text] with [Formula: see text] are realizable as the clique-to-vertex radius, clique-to-vertex monophonic radius, and clique-to-vertex detour radius of some connected graph and also it is shown that any three positive integers [Formula: see text] with [Formula: see text] are realizable as the clique-to-vertex diameter, clique-to-vertex monophonic diameter, and clique-to-vertex detour diameter of some connected graph. We introduce the clique-to-vertex monophonic center [Formula: see text] and the clique-to-vertex monophonic periphery [Formula: see text] and it is shown that the clique-to-vertex monophonic center does not lie in a single block of [Formula: see text].