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Recognizing connectedness from vertex-deleted subgraphs

Bowler, Andrew and Brown, Paul and Fenner, Trevor and Myrvold, W. (2010) Recognizing connectedness from vertex-deleted subgraphs. Journal of Graph Theory 67 (4), pp. 285-299. ISSN 0364-9024.

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Official URL: http://dx.doi.org/10.1002/jgt.20531

Abstract

Let G be a graph of order n. The vertex-deleted subgraph G − v, obtained from G by deleting the vertex v and all edges incident to v, is called a card of G. Let H be another graph of order n, disjoint from G. Then the number of common cards of G and H is the maximum number of disjoint pairs (v, w), where v and w are vertices of G and H, respectively, such that G − v≅H − w. We prove that if G is connected and H is disconnected, then the number of common cards of G and H is at most ⌊n/2⌋ + 1. Thus, we can recognize the connectedness of a graph from any ⌊n/2⌋ + 2 of its cards. Moreover, we completely characterize those pairs of graphs that attain the upper bound and show that, with the exception of six pairs of graphs of order at most 7, any pair of graphs that attains the maximum is in one of four infinite families.

Item Type: Article
Keyword(s) / Subject(s): reconstruction conjecture, reconstruction number, card, vertex-deleted subgraph
School or Research Centre: Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Computer Science and Information Systems
Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
Depositing User: Administrator
Date Deposited: 17 Aug 2011 11:21
Last Modified: 17 Apr 2013 12:21
URI: http://eprints.bbk.ac.uk/id/eprint/4053

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