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

TL;DR

This work tackles the Reconstruction Conjecture by presenting a reduction anchored in domination-number and diameter considerations, showing RC is equivalent to reconstructibility for two critical graph classes. It introduces two new pair-vertex parameters, $dv(G,k_1,k_2,k_3)$ and $dav(G,k_1,k_2,k_3)$, and proves these can be reconstructed from the deck for connected graphs with $\gamma(G)\ge 3$ via recursive relations, enabling recovery of vertex-pair degrees. For graphs with $\gamma(G)=2$, the authors establish recognizability from the deck by classifying graphs according to the diameter of the complement and leveraging $pv$/$pav$ parameters. They further analyze $k$-geodetic graphs of diameter two, identifying a subtree of graphs where reconstruction is feasible under the proposed framework. Overall, the paper integrates domination-based reductions with new pair-structure parameters to advance both recognition and reconstruction facets of RC, outlining steps toward a complete resolution.

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 $γ(G)=2$ 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 $γ(G)=2$ 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 $γ(G)\geq 3$, 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 $γ(G),γ(\overline{G})\geq 3$ are reconstructible under conditions where a vertex degree matches the size of a specific subset derived from these parameters.

Figures (1)

• Figure 1: Structure of a graph $G \in \mathscr{H}$ and its complement

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Theorem 5
• proof
• Theorem 6
• Theorem 7
• proof