Brown, Paul and Fenner, Trevor (2018) A new approach to graph reconstruction using supercards. Journal of Combinatorics 9 (1), pp. 95-118. ISSN 2156-3527.
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Abstract
The vertex-deleted subgraph G - v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex- deleted subgraphs. The number of common cards of G and H is the cardinality of a maximum multiset of common cards, i.e., the multiset intersection of the decks of G and H. We introduce a new approach to the study of common cards using supercards, where we define a supercard G+ of G and H to be a graph that has at least one vertex-deleted subgraph isomorphic to G, and at least one isomorphic to H. We show how maximum sets of common cards of G and H correspond to certain sets of permutations of the vertices of a supercard, which we call maximum saturating sets. We then show how to construct supercards of various pairs of graphs for which there exists some maximum saturating set X contained in Aut(G+). For certain other pairs of graphs, we show that it is possible to construct G+ and a maximum saturating set X such that the elements of X that are not in Aut(G+) are in one- to-one correspondence with a set of automorphisms of a different supercard G+_lambda � of G and H. Our constructions cover nearly all of the published families of pairs of graphs that have a large number of common cards.
Metadata
| Item Type: | Article |
|---|---|
| Keyword(s) / Subject(s): | Graph reconstruction, reconstruction numbers, vertex-deleted sub- graphs, supercards, graph automorphisms |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Trevor Fenner |
| Date Deposited: | 08 May 2017 13:55 |
| Last Modified: | 09 Aug 2023 12:41 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/18666 |
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