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Computational Complexity in Graph Theory
Jelínková, Eva
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 thesis, we study the computational complexity the problem S(P) for a certain graph property P: given a graph G, determine if G is switching-equivalent to a graph having P. First, we give an overview of known results, including both properties P for which S(P) is polynomial, and those for which S(P) is NP-complete. Then we show the NP-completeness of the following problem for each c (0; 1): determine if a graph G can be switched to contain a clique of size at least cn, where n is the number of vertices of G. We also study the problem if, for a xed graph H, a given graph is switching-equivalent to an H-free graph. We show that for H isomorphic to a claw, the problem is polynomial. Further, we give a characterization of graphs witching-equivalent to a K1;2-free graph by ten forbidden induced subgraphs, each having ve vertices.
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