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    A new approach to graph reconstruction using supercards

    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 Schools and Departments > School of Business, Economics & Informatics > Computer Science and Information Systems
    Depositing User: Trevor Fenner
    Date Deposited: 08 May 2017 13:55
    Last Modified: 27 Jul 2019 22:03
    URI: http://eprints.bbk.ac.uk/id/eprint/18666

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