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Keywords: distance-k vertex cover, H-free graph, np-completeness, polynomial-time algorithm, dichotomy
Published in RUP: 16.07.2021; Views: 1370; Downloads: 30
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Keywords: symmetric graph, non-Cayley graph, regular cover
Published in RUP: 18.11.2019; Views: 2145; Downloads: 99
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Abstract: Let ▫$\wp \colon \tilde{X} \to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$\rm{CT}_\wp$▫ being abelian. Assuming that a group of automorphisms ▫$G \le \rm{Aut} X$▫ lifts along $\wp$ to a group ▫$\tilde{G} \le \rm{Aut} \tilde{X}$▫, the problem whether the corresponding exact sequence ▫$\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$\tilde{G}$▫ nor the action ▫$G\to \rm{Aut} \rm{CT}_\wp$▫ nor a 2-cocycle ▫$G \times G \to \rm{CT}_\wp$▫, are given. Explicitly constructing the cover ▫$\tilde{X}$▫ together with ▫$\rm{CT}_\wp$▫ and ▫$\tilde{G}$▫ as permutation groups on ▫$\tilde{X}$▫ is time and space consuming whenever ▫$\rm{CT}_\wp$▫ is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$\rm{CT}_\wp$▫ is elementary abelian.
Keywords: algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product
Published in RUP: 15.10.2015; Views: 2773; Downloads: 157
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