On Graphs and Codes Preserved by Edge Local Complementation

TL;DR

This paper introduces ELC-preserved graphs, characterizes their properties, and explores their connection to error-correcting codes, providing classifications, constructions, and equivalence results for these graph classes.

Contribution

It defines ELC-preserved graphs, classifies small instances, provides recursive constructions, and links these graphs to binary linear codes, advancing understanding of graph-code relationships.

Findings

All ELC-preserved graphs up to order 12 found

Infinite families of ELC-preserved graphs constructed recursively

ELC-preserved graphs relate to Hamming codes and LC equivalence

Abstract

Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new graph class, ELC-preserved graphs, comprising all graphs that have an ELC orbit of size one. Through an exhaustive search, we find all ELC-preserved graphs of order up to 12 and all ELC-preserved bipartite graphs of order up to 16. We provide general recursive constructions for infinite families of ELC-preserved graphs, and show that all known ELC-preserved graphs arise from these constructions or can be obtained from Hamming codes. We also prove that certain pairs of ELC-preserved graphs are LC equivalent. We define ELC-preserved codes as binary linear codes corresponding to bipartite ELC-preserved graphs, and study the parameters of such codes.