### Abstract

A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter riot equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs.

Original language | English (US) |
---|---|

Pages (from-to) | 353-366 |

Number of pages | 14 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2002 |

### Fingerprint

### Keywords

- Algorithms
- Asteroidal triple-free graphs
- Dominating pair graphs
- Graph classes

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*15*(3), 353-366. https://doi.org/10.1137/S0895480100367111

**Dominating pair graphs.** / Deogun, Jitender S; Kratsch, Dieter.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 15, no. 3, pp. 353-366. https://doi.org/10.1137/S0895480100367111

}

TY - JOUR

T1 - Dominating pair graphs

AU - Deogun, Jitender S

AU - Kratsch, Dieter

PY - 2002/5/1

Y1 - 2002/5/1

N2 - A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter riot equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs.

AB - A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter riot equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs.

KW - Algorithms

KW - Asteroidal triple-free graphs

KW - Dominating pair graphs

KW - Graph classes

UR - http://www.scopus.com/inward/record.url?scp=10644223543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10644223543&partnerID=8YFLogxK

U2 - 10.1137/S0895480100367111

DO - 10.1137/S0895480100367111

M3 - Article

VL - 15

SP - 353

EP - 366

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -