Switching Reconstruction of Digraphs

TL;DR

This paper investigates the reconstructibility of digraphs through vertex switching, identifying all non-reconstructible cases with maximum degree 2 and characterizing switching-stable graphs.

Contribution

It precisely classifies non-reconstructible oriented graphs with degree at most 2 and fully characterizes switching-stable oriented graphs.

Findings

44 non-reconstructible oriented graphs with max degree 2

Complete set of switching-stable oriented graphs identified

Advances understanding of digraph reconstructibility

Abstract

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original.