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 d _min and d _max such that each leaf l _u of T corresponds to a vertex u∈V and there is an edge (u, v)∈E if and only if d _min ≤ d _T,w (l _u , l _v ) ≤ d _max , where d _T,w (l _u , l _v ) is the sum of the weights of the edges on the unique path from l _u to l _v 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 W _n , 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 PCG admits a full binary tree as witness tree T.