Pairwise Compatibility Graphs of Caterpillars

A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dmin and dmax such that each leaf lu of T corresponds to a vertex u of V and there is an edge (u, v) in E if and only if dmin <= dT,w(lu, lv) <= dmax where dT,w(lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T.  In this paper, we focus our attention on PCGs for which the witness tree is a caterpillar.  We first give some properties of graphs that are PCGs of a caterpillar. We formulate this problem as an integer linear programming problem and we exploit this formulation to show that  for the wheels on n vertices Wn, n = 7, ... , 11, the witness tree cannot be a caterpillar. Related to this result, we conjecture that no wheel is PCG of a caterpillar.  Finally, we state a more general result proving that any pairwise compatibility graph admits a full binary tree as witness tree T.