Hardness and approximation for L-EPG and B1-EPG graphs

Dror Epstein; Martin Charles Golumbic; Abhiruk Lahiri; Gila Morgenstern

doi:10.1016/j.dam.2019.10.004

Hardness and approximation for L-EPG and B₁-EPG graphs

Dror Epstein, Martin Charles Golumbic, Abhiruk Lahiri, Gila Morgenstern

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths and its subclass of ⌞-shaped paths, namely, the classes known as B₁-EPG and ⌞-EPG graphs, respectively. We show that fully-subdivided graphs are ⌞-EPG graphs, and later use this result in order to show that the following problems are APX-hard on ⌞-EPG graphs: MINIMUM VERTEX COVER, MAXIMUM INDEPENDENT SET, and MAXIMUM WEIGHTED INDEPENDENT SET, and also that MINIMUM DOMINATING SET is NP-complete on ⌞-EPG graphs. We also observe that MINIMUM COLORING is NP-complete already on ⌞-EPG, which follows from a proof for B₁-EPG in Epstein et al. (2013). Finally, we provide efficient constant-factor-approximation algorithms for each of these problems on B₁-EPG graphs.

Original language	English
Pages (from-to)	224-228
Number of pages	5
Journal	Discrete Applied Mathematics
Volume	281
DOIs	https://doi.org/10.1016/j.dam.2019.10.004
State	Published - 15 Jul 2020

Bibliographical note

Funding Information:
The authors thank Therese Biedl for her advice and insight which lead to a much stronger paper. The third author would also like to thank Manu Basavaraju and Mathew Francis for many helpful discussions. The authors wish to thank the School of Computer Science and Engineering at the Hebrew University of Jerusalem, where the second author is a visiting professor, for providing its facilities for our research activities.

Publisher Copyright:
© 2019 Elsevier B.V.

Keywords

B1-EPG graphs
Edge intersection graph
Fully-subdivided graphs
Paths on a grid
k-simplicial graphs

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.1016/j.dam.2019.10.004

Cite this

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title = "Hardness and approximation for L-EPG and B1-EPG graphs",

abstract = "The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths and its subclass of ⌞-shaped paths, namely, the classes known as B1-EPG and ⌞-EPG graphs, respectively. We show that fully-subdivided graphs are ⌞-EPG graphs, and later use this result in order to show that the following problems are APX-hard on ⌞-EPG graphs: MINIMUM VERTEX COVER, MAXIMUM INDEPENDENT SET, and MAXIMUM WEIGHTED INDEPENDENT SET, and also that MINIMUM DOMINATING SET is NP-complete on ⌞-EPG graphs. We also observe that MINIMUM COLORING is NP-complete already on ⌞-EPG, which follows from a proof for B1-EPG in Epstein et al. (2013). Finally, we provide efficient constant-factor-approximation algorithms for each of these problems on B1-EPG graphs.",

keywords = "B1-EPG graphs, Edge intersection graph, Fully-subdivided graphs, Paths on a grid, k-simplicial graphs",

author = "Dror Epstein and Golumbic, {Martin Charles} and Abhiruk Lahiri and Gila Morgenstern",

note = "Funding Information: The authors thank Therese Biedl for her advice and insight which lead to a much stronger paper. The third author would also like to thank Manu Basavaraju and Mathew Francis for many helpful discussions. The authors wish to thank the School of Computer Science and Engineering at the Hebrew University of Jerusalem, where the second author is a visiting professor, for providing its facilities for our research activities. Publisher Copyright: {\textcopyright} 2019 Elsevier B.V.",

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N1 - Funding Information: The authors thank Therese Biedl for her advice and insight which lead to a much stronger paper. The third author would also like to thank Manu Basavaraju and Mathew Francis for many helpful discussions. The authors wish to thank the School of Computer Science and Engineering at the Hebrew University of Jerusalem, where the second author is a visiting professor, for providing its facilities for our research activities. Publisher Copyright: © 2019 Elsevier B.V.

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