On a connection between the switching separability of a graph and that of its subgraphs

TL;DR

This paper explores the relationship between a graph's switching separability and that of its subgraphs, establishing conditions under which the property is preserved or not, and linking it to Boolean functions and quasigroup reducibility.

Contribution

It proves that if removing one or two vertices always yields a switching separable subgraph, then the original graph is also switching separable, and constructs counterexamples for odd orders greater than 4.

Findings

Graphs with certain vertex removal properties are switching separable

Counterexamples exist for odd order graphs greater than 4

Connections are established with Boolean function separability and quasigroup reducibility

Abstract

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: if removing any one or two vertices of a graph always results in a switching separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order greater than 4, there is a graph that is not switching separable, but removing any vertex always results in a switching separable subgraph. We show a connection with similar facts on the separability of Boolean functions and reducibility of $n$-ary quasigroups. Keywords: two-graph, reducibility, separability, graph switching, Seidel switching, graph connectivity, $n$-ary quasigroup