Reconstruction of graphs via asymmetry

TL;DR

The paper introduces a method for reconstructing graphs by identifying unique subgraphs called anchors, which leverage asymmetry to simplify the reconstruction process, proving trees are reconstructible.

Contribution

It proposes using anchors based on graph asymmetry to reduce the graph reconstruction problem to smaller, manageable subproblems.

Findings

Anchors are effective in graph reconstruction.

Trees are shown to be reconstructible using anchors.

Asymmetry-based anchors simplify the reconstruction process.

Abstract

Any graph which is not vertex transitive has a proper induced subgraph which is unique due to its structure or the way of its connection to the rest of the graph. We have called such subgraph as an anchor. Using an anchor which, in fact, is representative of a graph asymmetry, the reconstruction of that graph reduces to a smaller form of the reconstruction. Therefore, to show that a graph is reconstructible, it is sufficient to find a suitable anchor that brings us to a solved form of the problem. An orbit O of a graph G which makes G\ O to be an anchor or two vertices which makes G \{v,w} to be an anchor with the conditions that will be mentioned, is sufficient to show that G is reconstructible. For instance, this fact is enough to show that trees are reconstructible.