  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.
