Abstract
Graphs and Algorithms 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.
Highlights
The concept of Seidel’s switching was introduced by a Dutch mathematician J
We determine the fixed-parameter complexity of several closely related variants: we prove that switching to a graph of maximum degree at most k is fixed parameter tractable, as well as of switching to a graph of minimum degree at least k, and of switching to a k-regular graph
We summarize our results in the following theorem: Theorem 3.8 The problem SWITCH-k-SMALL-DEGS is linear fixed-parameter tractable — it can be solved in time O(4.614k · n + m)
Summary
Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic 3Cluster of Excellence “Multimodal Computing and Interaction”, Saarland University, Saarbrucken, Germany 4Faculty of Informatics, Masaryk University, Brno, Czech Republic. 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. 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 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.
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