### Abstract

A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a “ground line”, a line with slope -1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G’s bipartite adjacency matrix with rows A and columns B excludes three small ‘forbidden’ submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A|^{3}|B|^{3} -time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.

Original language | English (US) |
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Title of host publication | Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Proceedings |

Editors | Therese Biedl, Andreas Kerren |

Publisher | Springer-Verlag |

Pages | 303-316 |

Number of pages | 14 |

ISBN (Print) | 9783030044138 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 26th International Symposium on Graph Drawing and Network Visualization, GD 2018 - Barcelona, Spain Duration: Sep 26 2018 → Sep 28 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11282 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 26th International Symposium on Graph Drawing and Network Visualization, GD 2018 |
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Country | Spain |

City | Barcelona |

Period | 9/26/18 → 9/28/18 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Proceedings*(pp. 303-316). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11282 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-030-04414-5_21