Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond

TL;DR

This paper explores convexity properties in special graph classes, providing new algorithms and resolving a longstanding conjecture about the relationship between Steiner and geodetic numbers in AT-free graphs.

Contribution

It introduces a linear-time algorithm for the geodetic number in tree-cographs, proves the Steiner number is at least the geodetic number in AT-free graphs, and offers polynomial algorithms for related problems.

Findings

Linear-time algorithm for geodetic number in tree-cographs

Steiner number ≥ geodetic number in AT-free graphs

NP-completeness of maximal monophonic set in AT-free graphs

Abstract

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in $\AT$-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in $P_4$- sparse graphs.