A topology τ on the vertices of a comparability graph G is
said to be compatible with G if each subgraph H of G
is graph-connected if and only if it is a connected subspace of
(G,τ). In two previous papers we considered the problem of
finding compatible topologies for a given comparability graph and
we proved that the nonexistence of infinite paths was a necessary
and sufficient condition for the existence of a compact compatible
topology on a tree (that is to say, a connected graph without
cycles) and we asked whether this condition characterized the
existence of a compact compatible topology on a comparability
graph in which all cycles are of length at most n. Here we prove
an extension of the above-mentioned theorem to graphs whose cycles
are all of length at most five and we show that this is the best
possible result by exhibiting a comparability graph in which all
cycles are of length 6, with no infinite paths, but which has no
compact compatible topology.
