Unique square property and fundamental factorizations of graph bundles

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Abstract

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 finds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation δ* 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 define 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. © 2002 Elsevier Science B.V. All rights reserved.

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Zmazek, B., & Žerovnik, J. (2002). Unique square property and fundamental factorizations of graph bundles. Discrete Mathematics, 244(1–3), 551–561. https://doi.org/10.1016/S0012-365X(01)00106-6

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