On identity Seidel switches

TL;DR

This paper investigates the properties and structure of identity Seidel switches, which are vertex subsets that leave a graph invariant up to isomorphism, revealing their algebraic structure and constraints on graph configurations.

Contribution

It introduces the concept of identity Seidel switches, analyzes their group structure, and provides structural constraints and characterizations for graphs with many such switches.

Findings

Identity Seidel switches form an abelian 2-group under composition.

Necessary degree conditions for graphs with many identity switches.

Characterization of certain edge-identity switches via automorphisms.

Abstract

Seidel switching is a classical operation on graphs which plays a central role in the theory of two-graphs, signed graphs, and switching classes. In this paper we focus on those switches which leave a given graph invariant up to isomorphism. We call such subsets of the vertex set \emph{identity Seidel switches}. After recalling basic properties of Seidel switching and the associated abelian group structure, we introduce Seidel equivalence classes of graphs and then study the structure of the family of identity Seidel switches of a fixed graph. We show that this family forms a 14 pages; 2--group under composition, and we obtain structural constraints on graphs in which many vertices or edges give rise to identity switches. In particular, we derive necessary conditions in terms of degree parameters, and we characterize certain edge-identity switches via an automorphism of an induced subgraph. Several constructions and examples are presented, and some open problems are proposed.