### Abstract

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n−k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCArepresentations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 1-13 |

Number of pages | 13 |

Volume | 9214 |

ISBN (Print) | 9783319218397 |

DOIs | |

State | Published - 2015 |

Event | 14th International Symposium on Algorithms and Data Structures, WADS 2015 - Victoria, Canada Duration: Aug 5 2015 → Aug 7 2015 |

### 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 | 9214 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 14th International Symposium on Algorithms and Data Structures, WADS 2015 |
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Country | Canada |

City | Victoria |

Period | 8/5/15 → 8/7/15 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 9214, pp. 1-13). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9214). Springer Verlag. https://doi.org/10.1007/978-3-319-21840-3_1

**Contact graphs of circular arcs.** / Alam, Md Jawaherul; Eppstein, David; Kaufmann, Michael; Kobourov, Stephen G; Pupyrev, Sergey; Schulz, André; Ueckerdt, Torsten.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 9214, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9214, Springer Verlag, pp. 1-13, 14th International Symposium on Algorithms and Data Structures, WADS 2015, Victoria, Canada, 8/5/15. https://doi.org/10.1007/978-3-319-21840-3_1

}

TY - GEN

T1 - Contact graphs of circular arcs

AU - Alam, Md Jawaherul

AU - Eppstein, David

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Pupyrev, Sergey

AU - Schulz, André

AU - Ueckerdt, Torsten

PY - 2015

Y1 - 2015

N2 - We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n−k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCArepresentations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

AB - We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n−k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCArepresentations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

UR - http://www.scopus.com/inward/record.url?scp=84951855468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951855468&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-21840-3_1

DO - 10.1007/978-3-319-21840-3_1

M3 - Conference contribution

AN - SCOPUS:84951855468

SN - 9783319218397

VL - 9214

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 13

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -