### Abstract

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 C_{t+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 language | English |
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Pages (from-to) | 849-853 |

Number of pages | 5 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - Dec 1 2008 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)