On the hardness of switching to a small number of edges

TL;DR

This paper proves that deciding whether a graph can be switched to have at most a certain number of edges is NP-complete, even for graphs with bounded density, correcting a previous flawed proof.

Contribution

It provides a correct proof of NP-completeness for the switching problem and extends the result to graphs with fixed density bounds.

Findings

NP-completeness of switching to small number of edges established

NP-completeness holds for graphs with bounded density

Corrects previous flawed proof by Jelínková et al.

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].