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Reconstructing edge-deleted unicyclic graphs

Anthony E. Pizzimenti, Umarkhon Rakhimov

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, Müller verified the conjecture for graphs with $n$ vertices and $n \log_2(n)$ edges, improving on Lovás'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.

Reconstructing edge-deleted 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, Müller verified the conjecture for graphs with vertices and edges, improving on Lovás's bound of . Here, we show that the reconstruction conjecture holds for graphs which have exactly one cycle and and three non-isomorphic subtrees.

Paper Structure

This paper contains 4 sections, 3 theorems, 2 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Edge reconstruction
  3. Lemmas
  4. Theorem

Key Result

Lemma 1

Suppose that $G=(V,E)$ is unicyclic, and set $p \coloneqq ucd(G)$. Then $G_i = (V,E \setminus \{e_i\})$ for $e_i \in E(G)$ is one of the following: for any $i=1,\ldots,m$, where $m=|E|$.

Figures (2)

  • Figure 1:
  • Figure 2: The unicyclic, one-edge-deleted subgraphs $\mathcal{U} = \{U_1, \dots, U_5\}$ of $G$.

Theorems & Definitions (9)

  • Conjecture : Harary
  • Definition 2.1
  • Lemma 1
  • proof
  • Definition 2.2: Isomorphic as rooted trees, unique branch
  • Lemma 2
  • proof
  • Theorem 3
  • proof