On the Hardness of Switching to a Small Number of Edges
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].
KeywordsSeidel’s switching Computational complexity Graph density Switching-minimal graphs NP-completeness
- 4.Hage, J.: Structural Aspects of Switching Classes. Ph.D. thesis, Leiden Institute of Advanced Computer Science (2001)Google Scholar