COCOON 2016: Computing and Combinatorics pp 159-170

# On the Hardness of Switching to a Small Number of Edges

• Vít Jelínek
• Eva Jelínková
• Jan Kratochvíl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

## 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.

Jelínková 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 Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].

## Keywords

Seidel’s switching Computational complexity Graph density Switching-minimal graphs NP-completeness

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Vít Jelínek
• 1
• Eva Jelínková
• 2
Email author
• Jan Kratochvíl
• 2
1. 1.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic
2. 2.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic