### Parameterized Problems Related to Seidel's Switching

*Eva Jelinkova, Ondrej Suchy, Petr Hlineny, Jan Kratochvil*

#### 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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