Undirected Graphs of Entanglement 3

Entanglement is a complexity measure of digraphs that origins in fixedpoint 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 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.