Iterated clique graphs and bordered compact surfaces

The clique graph K(G) of a graph G is the intersection graph of all its (maximal) cliques. A graph G is said to be K-divergent if the sequence of orders of its iterated clique graphs |Kn(G)| tends to infinity with n, otherwise it is K-convergent. K-divergence is not known to be computable and there is even a graph on 8 vertices whose K-behavior is unknown. It has been shown that a regular Whitney triangulation of a closed surface is K-divergent if and only if the Euler characteristic of the surface is non-negative. Following this remarkable result, we explore here the existence of K-convergent and K-divergent (Whitney) triangulations of compact surfaces and find out that they do exist in all cases except (perhaps) where previously existing conjectures apply: it was conjectured that there is no K-divergent triangulation of the disk, and that there are no K-convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle. Our results seem to suggest that the topology still determines the K-behavior in these cases.