Hereditary domination in graphs: Characterization with forbidden induced subgraphs

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The leaf graph of a connected graph is obtained by joining a new vertex of degree one to each noncutting vertex. We prove that if a connected graph G is not dominated by any of its induced paths, then G is dominated by a connected induced subgraph whose leaf graph, too, is an induced subgraph of G. It follows that, for every nonempty class V of connected graphs, all of the minimal graphs not dominated by any induced subgraph isomorphic to some D ∈ V are cycles (of well-determined lengths) and leaf graphs of some graphs H D. In particular, if V is closed under the operation of taking connected induced subgraphs, then the hereditarily D-dominated graphs are characterized by the following family of forbidden induced subgraphs: leaf graphs of the connected graphs that are not in D, but all of their connected induced subgraphs are in D, and the cycle Ct+2, where t is the length of the shortest path not in D (if D does not contain all paths). This solves a problem that was open since the 1980s. A solution for the case of induced-hereditary classes D has been found simultaneously by Bacsó by applying a different method.

Original languageEnglish
Pages (from-to)849-853
Number of pages5
JournalSIAM Journal on Discrete Mathematics
Issue number3
Publication statusPublished - Dec 1 2008


  • Dominating set
  • Forbidden induced subgraph
  • Induced-hereditary property
  • Leaf graph
  • Structural domination

ASJC Scopus subject areas

  • Mathematics(all)

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