Homomorphisms of signed graphs: An update

TL;DR

This paper updates the theory of homomorphisms in signed graphs, providing characterizations, algorithms, and exploring their relation to 2-edge-colored graphs, with implications for graph coloring and structural properties.

Contribution

It offers a new characterization of negative walk sets, an algorithm for related decision problems, and connects signed graph homomorphisms to 2-edge-colored graphs.

Findings

Characterization of negative walk sets in graphs

An easy algorithm for the decision problem

Relations between signed graph homomorphisms and 2-edge-colored graphs

Abstract

A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph $(G,\sigma)$ to a signed graph $(H, \pi)$ to be a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ which preserves incidence, adjacency and the signs of closed walks. In this work we first give a characterization of the sets of closed walks in a graph $G$ that correspond to the set of negative walks in some signed graph on $G$. We also give an easy algorithm for the corresponding decision problem. After verifying the equivalence between this definition and earlier ones, we discuss the relation between homomorphisms of signed graphs and those of 2-edge-colored graphs. Next we provide some basic no-homomorphism lemmas. These lemmas lead to a general method of defining chromatic number which is discussed at length. Finally, we list a few problems that are the driving force behind the study of homomorphisms of signed graphs.