A Reduction of the Reconstruction Conjecture using Domination and Vertex Pair Parameters

TL;DR

This paper advances the graph reconstruction conjecture by proving reconstructibility for specific graph classes using new parameters and reductions, focusing on graphs with domination number 2 and diameter conditions.

Contribution

It introduces new parameters for graph reconstruction, establishes a reduction of the RC to certain 2-connected graphs, and proves reconstructibility for graphs with specific domination and diameter properties.

Findings

Graphs with domination number 2 are recognizable from their decks.

Reconstruction of certain 2-connected graphs is equivalent to the RC.

New parameters are reconstructible for large connected graphs with domination number ≥ 3.

Abstract

A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex-deleted subgraphs, known as the deck of G. The Reconstruction Conjecture (RC) posits that every finite simple graph with at least three vertices is reconstructible. In this paper, we prove that the class of graphs with domination number $\gamma(G)=2$ is recognizable from the deck $D(G)$. We also establish a new reduction of the RC: it holds if and only if all $2$-connected graphs $G$ with $\gamma(G)=2$ or $\operatorname{diam}(G)=\operatorname{diam}(\overline{G})=2$ are reconstructible. To aid reconstruction, we introduce two new parameters: $dv(G,k_1,k_2,k_3)$, which counts the number of non-adjacent vertex pairs in $G$ with $k_1$ common neighbours, $k_2$ neighbours exclusive to the first vertex, and $k_3$ exclusive to the second; and $dav(G,k_1,k_2,k_3)$, defined analogously for adjacent pairs. For connected graphs with at least $12$ vertices and $\gamma(G)\geq 3$, we show these parameters are reconstructible from $D(G)$ via recursive equations and induction. Finally, we prove that $k$-geodetic graphs of diameter two with $\gamma(G),\gamma(\overline{G})\geq 3$ are reconstructible under conditions where a vertex degree matches the size of a specific subset derived from these parameters.