Pairwise Compatibility Graphs of Caterpillars

A graph G 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 ∈ V and there is an edge (u, v) ∈ 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 concentrate 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. Then, we reformulate 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 turn our attention to PCGs of a general tree and prove that all of them admit as witness tree T a full binary tree.