Monophonic position sets of Cartesian and lexicographic products of graphs

TL;DR

This paper investigates the monophonic position number in Cartesian and lexicographic product graphs, providing bounds, structural characterizations, and formulas that extend understanding of induced path configurations in complex graph products.

Contribution

It introduces a detailed structural analysis of monophonic position sets in Cartesian products and derives a formula for lexicographic products based on clique number and factor structures.

Findings

Monophonic position sets in Cartesian products are classified into three canonical forms.

Bounds for the monophonic position number in Cartesian products are established, coinciding under certain conditions.

A formula for the monophonic position number of lexicographic products is provided, involving clique number and factor structures.

Abstract

The general position problem in graph theory asks for the number of vertices in a largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. The analogous monophonic position problem is obtained from the general position problem by replacing ``shortest path'' by ``induced path.'' In this paper the monophonic position number is studied on Cartesian and lexicographic products of graphs. It is proved that in Cartesian products, a monophonic position set can only be in one of three canonical forms, named layered, varied, and cliquey. The monophonic position number of an arbitrary Cartesian product is bounded from below and above. The two bounds coincide if neither of the factors has simplicial vertices. A formula for the monophonic position number of a lexicographic product is given which only contains the clique number and the structure of monophonic sets of the second factor.