A construction of the abstract induced subgraph poset of a graph from its abstract edge subgraph poset

TL;DR

This paper demonstrates how to reconstruct the abstract induced subgraph poset of a graph from its abstract edge-subgraph poset, extending classical graph reconstruction results to more complex structures.

Contribution

It introduces a method to construct the abstract induced subgraph poset from the edge-subgraph poset, broadening the scope of graph reconstruction theory.

Findings

Reconstruction is possible for graphs without isolated vertices.

The method generalizes classical vertex deck reconstruction.

Certain families of graphs are exceptions to the reconstruction.

Abstract

The abstract induced subgraph poset of a graph is the isomorphism class of the induced subgraph poset of the graph, suitably weighted by subgraph counting numbers. The abstract bond lattice and the abstract edge-subgraph poset are defined similarly by considering the lattice of subgraphs induced by connected partitions and the poset of edge-subgraphs, respectively. Continuing our development of graph reconstruction theory on these structures, we show that if a graph has no isolated vertices, then its abstract bond lattice and the abstract induced subgraph poset can be constructed from the abstract edge-subgraph poset except for the families of graphs that we characterise. The construction of the abstract induced subgraph poset from the abstract edge-subgraph poset generalises a well known result in reconstruction theory that states that the vertex deck of a graph with at least 4 edges and without isolated vertices can be constructed from its edge deck.12