Recognizing Cartesian graph bundles

Abstract

Graph bundles generalize the notion of covering graphs and graph products. In this paper we extend some of the methods for recognizing Cartesian product graphs to graph bundles. Two main notions are used. The first one is the well-known equivalence relation δ ★ defined on the edge-set of a graph. The second one is the concept of k-convex subgraphs. A subgraph H is k-convex in G, if for any two vertices x and y of distance d, d ⩽ k, each shortest path from x to y in G is contained entirely in H. The main result is an algorithm that finds a representation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over a triangle-free simple base. The problem of recognizing graph bundles over a base containing triangles remains open.