## Abstract

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, k≥0.We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs are shown to be equivalent to the bipartite B_{0}-VPG graphs. Chordal B_{0}-VPG graphs are shown to be exactly Strongly Chordal B_{0}-VPG graphs. We prove the strict containment of B_{0}-VPG and circle graphs into B_{1}-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B_{3}-VPG graphs.In the case of B_{0}-VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for Bk-VPG graphs, for k≥0, is NP-complete and give a 2-approximation algorithm for coloring B_{0}-VPG graphs. Furthermore, we prove that triangle-free B_{0}-VPG graphs are 4-colorable, and this is best possible.

Original language | English |
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Pages (from-to) | 141-146 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 37 |

Issue number | C |

DOIs | |

State | Published - 1 Aug 2011 |

### Bibliographical note

Funding Information:1 partially supported by the Israel Science Foundation grant 347/09

## Keywords

- Chordal Graphs
- Circle Graphs
- Graph Coloring
- Grid Intersection Graphs
- Helly Property
- Planar Graphs
- String Graphs
- Triangle-free Graphs

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics