Reconstructing edge-deleted unicyclic graphs

TL;DR

This paper proves the Harary reconstruction conjecture for a specific class of unicyclic graphs with exactly one cycle and three non-isomorphic subtrees, expanding the known cases where the conjecture holds.

Contribution

It establishes the conjecture for a new class of graphs characterized by a single cycle and three non-isomorphic subtrees, which was previously unverified.

Findings

Reconstruction conjecture holds for unicyclic graphs with one cycle and three non-isomorphic subtrees.

Extends the class of graphs for which the Harary reconstruction conjecture is proven.

Provides a new proof technique for reconstructing specific unicyclic graphs.

Abstract

The Harary reconstruction conjecture states that any graph with more than four edges can be uniquely reconstructed from its set of maximal edge-deleted subgraphs. In 1977, M\"uller verified the conjecture for graphs with $n$ vertices and $n \log_2(n)$ edges, improving on Lov\'as's bound of $\log(n^2-n)/4$. Here, we show that the reconstruction conjecture holds for graphs which have exactly one cycle and and three non-isomorphic subtrees.