Some computational aspects of solvable regular covers of graphs. (September 2015) - Požar, Rok
Abstract: Given a connected graph X and a group G of its automorphisms we first introduce an approach for constructing all pairwise nonequivalent connected solvable regular coverings ℘ : X ˜ → X (that is, with a solvable group of covering transformations CT ( ℘ ) ) along which G lifts, up to a prescribed order n of X ˜ . Next, given a connected solvable regular covering ℘ : X ˜ → X by means of voltages and a group G ≤ Aut ( X ) that lifts along ℘, we consider algorithms for testing whether the lifted group G ˜ is a split extension of CT ( ℘ ) . In computational group theory, methods for testing whether a given extension of permutation groups splits are known. However, in order to apply the existing algorithms, X ˜ together with CT ( ℘ ) and G ˜ need to be constructed in the first place, which is far from optimal. Recently, an algorithm avoiding such explicit constructions has been proposed by Malnič and Požar (submitted for publication) . We here provide additional details about this algorithm and investigate its performance compared to the one using explicit constructions. To this end, a concrete dataset of solvable regular covers of graphs has been generated by the algorithm mentioned in the first paragraph.