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 c1 of deletion:

(1) , , , and .

(2) For all k2, and .

(3) For all k2, .

(4) .

(5) For all k2, .

Similar to the definition of reconstruction numbers vrn(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451–454] and ern(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn(G) and ern(G), and give an example of a family {G_n}_n4 of graphs on n vertices for which vrn(G_n)<vrn(G_n). For every k2 and n1, we show that there exists a collection of k graphs on (2^k-1+1)n+k vertices with 2ⁿ 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.

Keywords: Graph reconstruction; Legitimate deck; Graph isomorphism; Reconstruction numbers

Article Outline

1. Introduction

1.1. Background

1.2. Our contributions

2. Preliminaries

2.1. Notation

2.2. Graph isomorphism

2.3. A tool for proving isomorphism between sets

2.4. Computational problems in graph reconstruction

3. Reconstruction from vertex and edge decks

3.1. Reconstruction from a complete deck

3.2. Reconstruction from a subdeck

3.2.1. Subdeck checking problems

3.2.2. Legitimate subdeck problems

4. Reconstruction numbers of graphs

5. Open problems

Acknowledgements

References