  Compatibility of unrooted phylogenetic trees is a well studied problem in
phylogenetics. It asks to determine whether for a set of k input trees there
exists a larger tree (called a supertree) that contains the topologies of all k
input trees. When any such supertree exists we call the instance compatible and
otherwise incompatible. It is known that the problem is NP-hard and FPT,
although a constructive FPT algorithm is not known. It has been shown that
whenever the treewidth of an auxiliary structure known as the display graph is
strictly larger than the number of input trees, the instance is incompatible.
Here we show that whenever the treewidth of the display graph is at most 2, the
instance is compatible. Furthermore, we give a polynomial-time algorithm to
construct a supertree in this case. Finally, we demonstrate both compatible and
incompatible instances that have display graphs with treewidth 3, highlighting
that the treewidth of the display graph is (on its own) not sufficient to
determine compatibility.
