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. 95118. ISSN 21563527.

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Abstract
The vertexdeleted 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 vertexdeleted 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 toone 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, vertexdeleted sub graphs, supercards, graph automorphisms 
School:  School of Business, Economics & Informatics > Computer Science and Information Systems 
Depositing User:  Trevor Fenner 
Date Deposited:  08 May 2017 13:55 
Last Modified:  09 Feb 2021 23:43 
URI:  https://eprints.bbk.ac.uk/id/eprint/18666 
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