### Abstract

The intersection dimension of a graph G with respect to a class A of graphs is the minimum k such that G is the intersection of at most k graphs on vertex set V(G) each of which belongs to A. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) if A is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect to A is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.: On grid intersection graphs, Discrete Math. 87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.

Original language | English |
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Pages (from-to) | 159-168 |

Number of pages | 10 |

Journal | Graphs and Combinatorics |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*10*(2), 159-168. https://doi.org/10.1007/BF02986660

**Intersection Dimensions of Graph Classes.** / Kratochvil, Jan; Tuza, Z.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 10, no. 2, pp. 159-168. https://doi.org/10.1007/BF02986660

}

TY - JOUR

T1 - Intersection Dimensions of Graph Classes

AU - Kratochvil, Jan

AU - Tuza, Z.

PY - 1994

Y1 - 1994

N2 - The intersection dimension of a graph G with respect to a class A of graphs is the minimum k such that G is the intersection of at most k graphs on vertex set V(G) each of which belongs to A. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) if A is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect to A is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.: On grid intersection graphs, Discrete Math. 87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.

AB - The intersection dimension of a graph G with respect to a class A of graphs is the minimum k such that G is the intersection of at most k graphs on vertex set V(G) each of which belongs to A. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) if A is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect to A is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.: On grid intersection graphs, Discrete Math. 87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.

UR - http://www.scopus.com/inward/record.url?scp=62849094501&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=62849094501&partnerID=8YFLogxK

U2 - 10.1007/BF02986660

DO - 10.1007/BF02986660

M3 - Article

AN - SCOPUS:62849094501

VL - 10

SP - 159

EP - 168

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -