Fully gated graphs: Recognition and convex operations

Charles Colbourn, Cécile Huybrechts

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

A graph is fully gated when every convex set of vertices is gated. Doignon posed the problem of characterizing fully gated graphs and in particular of deciding whether there is an efficient algorithm for their recognition. While the number of convex sets can be exponential, we establish that it suffices to examine only the convex hulls of pairs of vertices. This yields an elementary polynomial time algorithm for the recognition of fully gated graphs; however, it does not appear to lead to a simple structural characterization. In this direction, we establish that fully gated graphs are closed under a set of 'convex' operations, including a new operation which duplicates the vertices of a convex set (under some well-defined restrictions). This in turn establishes that every bipartite graph is an isometric subgraph of a fully gated graph, thereby severely limiting the potential for a characterization based on subgraphs. Finally, a large class of fully gated graphs is obtained using the presence of bipartite dominators, which suggests that simple convex operations cannot suffice to produce all fully gated graphs.

Original languageEnglish (US)
Pages (from-to)5184-5195
Number of pages12
JournalDiscrete Mathematics
Volume308
Issue number22
DOIs
StatePublished - Nov 28 2008

Keywords

  • Convex subgraph
  • Fully gated graph
  • Gated graph
  • Isometric embedding
  • Median graph

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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