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 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.