Edge Determining Sets and Determining Index

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.