Complexity Results in Graph Reconstruction

TL;DR

This paper explores the complexity of graph reconstruction problems and their relation to graph isomorphism, introducing new parameters and demonstrating their computational equivalences and differences.

Contribution

It establishes new complexity equivalences for graph reconstruction problems and introduces novel graph parameters with illustrative examples.

Findings

Graph isomorphism is equivalent to various graph reconstruction problems.

New graph parameters $vrn_{\forall}$ and $ern_{\forall}$ are introduced and analyzed.

Existence of graph families with exponentially many preimages is demonstrated.

Abstract

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts $c \geq 1$ of deletion: 1) $GI \equiv^{l}_{iso} VDC_{c}$, $GI \equiv^{l}_{iso} EDC_{c}$, $GI \leq^{l}_{m} LVD_c$, and $GI \equiv^{p}_{iso} LED_c$. 2) For all $k \geq 2$, $GI \equiv^{p}_{iso} k-VDC_c$ and $GI \equiv^{p}_{iso} k-EDC_c$. 3) For all $k \geq 2$, $GI \leq^{l}_{m} k-LVD_c$. 4)$GI \equiv^{p}_{iso} 2-LVC_c$. 5) For all $k \geq 2$, $GI \equiv^{p}_{iso} k-LED_c$. For many of these results, even the $c = 1$ case was not previously known. Similar to the definition of reconstruction numbers $vrn_{\exists}(G)$ [HP85] and $ern_{\exists}(G)$ (see page 120 of [LS03]), we introduce two new graph parameters, $vrn_{\forall}(G)$ and $ern_{\forall}(G)$, and give an example of a family $\{G_n\}_{n \geq 4}$ of graphs on $n$ vertices for which $vrn_{\exists}(G_n) < vrn_{\forall}(G_n)$. For every $k \geq 2$ and $n \geq 1$, we show that there exists a collection of $k$ graphs on $(2^{k-1}+1)n+k$ vertices with $2^{n}$ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.