Switching $(m, n)$-mixed graphs with respect to Abelian groups

TL;DR

This paper generalizes switching and homomorphism concepts from edge-colored graphs to $(m,n)$-mixed graphs with Abelian groups, establishing structural characterizations and complexity results including NP-hardness and NP-completeness.

Contribution

It extends previous results to Abelian groups, introduces a universal graph construction for switch equivalence, and analyzes the computational complexity of related homomorphism problems.

Findings

Existence of a universal $(m,n)$-mixed graph $P_ extGamma(H)$ for switch equivalence.

Deciding switchability to a non-core is NP-hard for arbitrary groups.

Switchable $k$-colouring problem exhibits a dichotomy for $m \\geq 1$.

Abstract

We extend results of Brewster and Graves for switching $m$-edge coloured graphs with respect to a cyclic group to switching $(m, n)$-mixed graphs with respect to an Abelian group. In particular, we establish the existence of a $(m, n)$-mixed graph $P_\Gamma(H)$ with the property that a $(m, n)$-mixed graph $G$ is switch equivalent to $H$ if and only if it is a special subgraph of $P_\Gamma(H)$, and the property that that $G$ can be switched to have a homomorphism to $H$ if and only if it has a homomorphism (without switching) to $P_\Gamma(H)$. We consider the question of deciding whether a $(m, n)$-mixed graph can be switched so that it has a homomorphism to a proper subgraph, i.e. whether it can be switched so that it isn't a core. We show that this question is NP-hard for arbitrary groups and NP-complete for Abelian groups. Finally, we consider the complexity of the switchable $k$-colouring problem for $(m, n)$-mixed graphs and prove a dichotomy theorem in the cases where $m \geq 1$.