On strong graph bundles

We study strong graph bundles : a concept imported from topology which generalizes both covering graphs and product graphs. Roughly speaking, a strong graph bundle always involves three graphs E, B and F and a projection p:E→B with fiber F (i.e. p−1x≅F for all x∈V(B)) such that the preimage of any edge xy of B is trivial (i.e. p−1xy≅K2⊠F). Here we develop a framework to study which subgraphs S of B have trivial preimages (i.e. p−1S≅S⊠F) and this allows us to compare and classify several variations of the concept of strong graph bundle. As an application, we show that the clique operator preserves triangular graph bundles (strong graph bundles where preimages of triangles are trivial) thus yielding a new technique for the study of clique divergence of graphs.