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Parameterized Problems Related to Seidel's Switching

Abstract : Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.
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Eva Jelinkova, Ondrej Suchy, Petr Hlineny, Jan Kratochvil. Parameterized Problems Related to Seidel's Switching. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 2 (2), pp.19--42. ⟨10.46298/dmtcs.542⟩. ⟨hal-00990491⟩

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