Undirected Graphs of Entanglement 3

TL;DR

This paper explores the structural properties of undirected graphs with entanglement at most 3, using Tutte's decomposition to identify necessary conditions for such graphs.

Contribution

It provides the first structural characterization of undirected graphs with entanglement 3 using Tutte's decomposition, extending previous partial results.

Findings

Necessary conditions on Tutte's tree for entanglement 3

Structural insights into 2-connected graphs of entanglement 3

Extension of entanglement characterization to undirected graphs

Abstract

Entanglement is a complexity measure of digraphs that origins in fixed-point logics. Its combinatorial purpose is to measure the nested depth of cycles in digraphs. We address the problem of characterizing the structure of graphs of entanglement at most $k$. Only partial results are known so far: digraphs for $k=1$, and undirected graphs for $k=2$. In this paper we investigate the structure of undirected graphs for $k=3$. Our main tool is the so-called \emph{Tutte's decomposition} of 2-connected graphs into cycles and 3-connected components into a tree-like fashion. We shall give necessary conditions on Tutte's tree to be a tree decomposition of a 2-connected graph of entanglement 3.