On a test on switching separability of graphs modulo $q$

TL;DR

This paper investigates the switching separability of weighted graphs modulo q, providing a test for separability, characterizing exceptions for even q, and linking it to the reducibility of certain algebraic structures.

Contribution

It proves a separability test for graphs modulo q, characterizes exceptions for even q, and connects graph separability to algebraic quasigroup reducibility.

Findings

The test holds for odd q.

Exceptions are characterized for even q.

A link to quasigroup reducibility is established.

Abstract

We consider the graphs whose edges are marked by the integers (weights) from $0$ to $q-1$ (zero corresponds to no-edge). Such graph is called additive if its vertices can be marked in such a way that the weight of every edge is equal to the modulo-$q$ sum of weights of the two incident vertices. By a switching of a graph we mean the modulo-$q$ sum of the graph with some additive graph on the same vertex set. A graph with $n$ vertices is called switching separable if some of its switchings does not have a connected component of order $n$ or $n-1$. We consider the following test for the switching separability: if removing any vertex of a graph $G$ results in a switching separable graph, then $G$ is switching separable itself. We prove this test for odd $q$ and characterize the exceptions when $q$ is even. We establish a connection between the switching separability of a graph and the reducibility of $(n-1)$-ary quasigroups constructed from this graph.