201. On the split structure of lifted groupsAleksander Malnič, Rok Požar, 2016, original scientific article 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: 2837; Downloads: 157 Full text (422,56 KB) |
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206. Linear separation of connected dominating sets in graphs : (extended abstract)Nina Chiarelli, Martin Milanič, 2014, published scientific conference contribution Keywords: povezana dominantna množica, hereditarni grafovski razred, dualno Spernerjev hipergraf, pragovni hipergraf, connected dominating set, hereditary graph class, dually Sperner hypergraph, treshold hypergraph Published in RUP: 15.10.2015; Views: 3584; Downloads: 114 Link to full text |
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208. On hereditary efficiently dominatable graphsMartin Milanič, 2011, published scientific conference contribution abstract Keywords: popolna koda, učinkovita dominacija, grafi z učinkovito dominantno množico, polinomski algoritmi, hereditarni grafovski razredi, perfect code, efficient domination, efficiently dominatable graphs, polynomial time algorithms, hereditary graph classes Published in RUP: 15.10.2015; Views: 3555; Downloads: 248 Link to full text |
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