Edge Determining Sets and Determining Index

TL;DR

This paper introduces the concept of an edge determining set and determining index in graphs, establishing bounds relating them to the traditional determining number, and explores their properties across various graph families.

Contribution

It defines the new concept of an edge determining set and determining index, proves bounds relating these to the determining number, and analyzes their properties and values for different graph classes.

Findings

Det'(G)  Det(G)  2Det'(G) for Det(G) eq 1

Bounds are sharp for infinite graph families

Determining index computed for several graph families

Abstract

A graph automorphism is a bijective mapping of the vertices that preserves adjacent vertices. A vertex determining set of a graph is a set of vertices such that the only automorphism that fixes those vertices is the identity. The size of a smallest such set is called the determining number, denoted Det$(G)$. The determining number is a parameter of the graph capturing its level of symmetry. We introduce the related concept of an edge determining set and determining index, Det$'(G)$. We prove that Det$'(G) \le \text{Det}(G) \le 2\text{Det}'(G)$ when Det$(G) \neq 1$ and show both bounds are sharp for infinite families of graphs. Further, we investigate properties of these new concepts, as well as provide the determining index for several families of graphs.