Pairwise compatibility graphs revisited

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
(l
u
, l
v
) ≤ d
max
where d
T
(l
u
, l
v
) is the sum of weights of the edges on the unique path from l
u
to l
v
in T. In this note, firstly, we show a class of bipartite graphs not to be PCG, that is, it is not possible to draw a pairwise compatibility tree for these graphs. Further we construct a class of more general non-PCGs each member of which contains a bipartite graph belonging to the class of graphs mentioned above as a subgraph. Secondly, we show by computational means that all the bipartite graphs with at most eight vertices are PCGs. In particular all these graphs are PCGs of a particular structure of tree called centipede.