Closure Under Minors of Undirected Entanglement

TL;DR

This paper proves that the class of undirected graphs with bounded entanglement is closed under minors, using a game-theoretic approach involving Robber and Cops games, advancing understanding of graph complexity measures.

Contribution

It establishes that undirected graphs with bounded entanglement are minor-closed, a property previously unknown for this complexity measure.

Findings

Bounded entanglement classes are minor-closed.

Game-theoretic characterization is key to the proof.

Advances understanding of graph complexity measures.

Abstract

Entanglement is a digraph complexity measure that origins in fixed-point theory. Its purpose is to count the nested depth of cycles in digraphs. In this paper we prove that the class of undirected graphs of entanglement at most $k$, for arbitrary fixed $k \in \mathbb{N}$, is closed under taking minors. Our proof relies on the game theoretic characterization of entanglement in terms of Robber and Cops games.