Cycle convexity and the tunnel number of links

TL;DR

This paper introduces a new graph convexity called Cycle Convexity, explores its properties, and studies the computational complexity of the associated hull number across different classes of graphs.

Contribution

The paper defines Cycle Convexity, analyzes its properties, and establishes complexity results for computing the hull number in various graph classes.

Findings

Hull number of 4-regular planar graphs is at most half of the vertices.

Computing the hull number for planar graphs is NP-complete.

Polynomial algorithms exist for chordal, P4-sparse graphs, and grids.

Abstract

In this work, we introduce a new graph convexity, that we call Cycle Convexity, motivated by related notions in Knot Theory. For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as $I_{cc}(S) = S\cup \{v\in V(G)\mid \text{there is a cycle }C\text{ in }G\text{ such that } V(C)\setminus S=\{v\}\}$, for every $S\subseteq V(G)$. We say that $S\subseteq V(G)$ is convex if $I_{cc}(S)=S$. The convex hull of $S\subseteq V(G)$, denoted by $Hull(S)$, is the inclusion-wise minimal convex set $S'$ such that $S\subseteq S'$. A set $S\subseteq V(G)$ is called a hull set if $Hull(S)=V(G)$. The hull number of $G$ in the cycle convexity, denoted by $hn_{cc}(G)$, is the cardinality of a smallest hull set of $G$. We first present the motivation for introducing such convexity and the study of its related hull number. Then, we prove that: the hull number of a 4-regular planar graph is at most half of its vertices; computing the hull number of a planar graph is an $NP$-complete problem; computing the hull humber of chordal graphs, $P_4$-sparse graphs and grids can be done in polynomial time.