Unique square property and fundamental factorizations of graph bundles

Graph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167=168 (1998) 393) authors constructed an algorithm that 5nds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation ∗ having the square property. An equivalence relation R on the edge set of a graph has the (unique) square property if and only if any pair of adjacent edges which belong to distinct R-equivalence classes span exactly one induced 4-cycle (with opposite edges in the same R-equivalence class). In this paper we de5ne the unique square property and show that any weakly 2-convex equivalence relation possessing the unique square property determines the fundamental factorization of a graph as a nontrivial cartesian graph bundle over an arbitrary base graph, whenever it separates degenerate and nondegenerate edges of the factorization. c © 2002 Elsevier Science B.V. All rights reserved.