On the hardness of switching to a small number of edges

Vít Jelínek; Eva Jelínková; Jan Kratochvíl

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arXiv · 1603.00254

On the hardness of switching to a small number of edges

Vít Jelínek·Eva Jelínková·Jan Kratochvíl

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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 one by a sequence of switches. Jel\'inkov\'a et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide if the input graph can be switched to contain at most a given number of edges. There turns out to be a flaw in their proof. We present a correct proof. Furthermore, we prove that the problem remains NP-complete even when restricted to graphs whose density is bounded from above by an arbitrary fixed constant. This partially answers a question of Matou\v{s}ek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].

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Vít Jelínek, Eva Jelínková, Jan Kratochvíl. 2016-03-01. On the hardness of switching to a small number of edges. https://arxiv.org/abs/1603.00254

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