A Simple Explanation for the Reconstruction of Graphs

TL;DR

This paper presents a simple, general framework that explains why certain graphs are reconstructible from their decks, focusing on structural properties like anchors and connectional anchors without proving the conjecture.

Contribution

It introduces a new conceptual framework based on anchors and connectional anchors to explain graph reconstructibility, providing insights into why some graphs are uniquely determined by their decks.

Findings

Non-regular graphs have unique induced subgraphs called anchors.

Graphs with specific orbit and removal conditions are reconstructible.

The framework explains reconstruction for small graphs verified by computer.

Abstract

The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of graphs. Here, we do not prove or reject the reconstruction conjecture. But, we explain why a graph is reconstructible. For instance, the reconstruction of small graphs which have been shown by computer, is explained in this framework. We show that any non-regular graph has a proper induced subgraph which is unique due to either its structure or the way of its connection to the rest of the graph. Here, the former subgraph is defined an anchor and the latter a connectional anchor, if it is distinguishable in the deck. We show that if a graph has an orbit with at least three vertices whose removal leaves an anchor, or it has two vertices whose removal leaves an anchor with the mentioned condition in the paper, then it is reconstructible. This simple statement can easily explain the reconstruction of a graph from its deck.