On the open monophonic number of a graph

Abstract

For a connected graph [Formula: see text] of order [Formula: see text], a connected open monophonic set [Formula: see text] of vertices in a connected graph [Formula: see text] is a minimal connected open monophonic set if no proper subset of [Formula: see text] is a connected open monophonic set of [Formula: see text]. The upper connected open monophonic number om[Formula: see text] of [Formula: see text] is the maximum cardinality of a minimal connected open monophonic set of [Formula: see text]. The upper connected open monophonic numbers of certain standard graphs are determined. It is proved that [Formula: see text] for the Petersen graph [Formula: see text]. Also, it is proved that for a graph [Formula: see text] of order [Formula: see text], [Formula: see text] if and only if [Formula: see text]. For positive integers [Formula: see text],[Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. Further, it is analyzed how the addition of a pendant edge to certain standard graphs affects the upper connected open monophonic number of the respective original graphs.