Complexity aspects of the triangle path convexity

TL;DR

This paper investigates the triangle path convexity in graphs, providing polynomial-time algorithms to compute the convexity and hull numbers, which measure the size of certain convex sets within the graph.

Contribution

It introduces the concepts of convexity and hull numbers in the triangle path convexity and develops efficient algorithms to compute them.

Findings

Polynomial algorithms for convexity number

Polynomial algorithms for hull number

Enhanced understanding of triangle path convexity properties

Abstract

A path $P = v_1, ..., v_t$ is a {\em triangle path} (respectively, {\em monophonic path}) of $G$ if no edges exist joining vertices $v_i$ and $v_j$ of $P$ such that $|j - i| > 2$; (respectively, $|j - i| > 1$). A set of vertices $S$ is {\em convex} in the triangle path convexity (respectively, monophonic convexity) of $G$ if the vertices of every triangle path (respectively, monophonic path) joining two vertices of $S$ are in $S$. The cardinality of a maximum proper convex set of $G$ is the {\em convexity number of $G$} and the cardinality of a minimum set of vertices whose convex hull is $V(G)$ is the {\em hull number of $G$}. Our main results are polynomial time algorithms for determining the convexity number and the hull number of a graph in the triangle path convexity.