On the vertex monophonic, vertex geodetic and vertex Steiner numbers of graphs

Abstract

Let [Formula: see text] be a vertex of a connected graph G. It is proved that [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-monophonic, [Formula: see text]-geodetic and [Formula: see text]-Steiner numbers of [Formula: see text], respectively. However, there is no relationship between [Formula: see text] and [Formula: see text]. It is proved that in distance-hereditary graphs, [Formula: see text]. It is shown that for any positive integers [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for some vertex [Formula: see text].