On monophonic position sets in graphs

TL;DR

This paper introduces the monophonic position problem in graph theory, analyzing its properties, bounds, and exact values for specific graph classes, and discusses its computational complexity.

Contribution

It defines and studies the monophonic position number, providing bounds, exact values for certain graph families, and exploring the problem's complexity.

Findings

Monophonic position number bounded by independence number in triangle-free graphs

Exact monophonic position numbers determined for unicyclic, bipartite complements, and split graphs

Complexity analysis of the monophonic position problem

Abstract

The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position problem called the \emph{monophonic position problem}, obtained by replacing `shortest path' by `induced path'. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.