### Abstract

In this paper we consider the problem of drawing a planar graph using circular-arcs as edges, given a one-to-one mapping between the vertices of the graph and a set of n points on the plane, where n is the number of vertices in the graph. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n ^{2}) algorithm to find out if a drawing with no crossings can be realized. We present an improved O(n ^{7/4} polylog n) time algorithm. For the special case where the possible circular arcs for each edge are of the same length, we present an even more efficient algorithm that runs in O(n ^{3/2} polylog n) time. We also consider the problem if we have more than two possible circular arcs per edge and show that the problem becomes NP-Hard. Moreover, we show that two optimization versions of the problem are also NP-Hard.

Original language | English (US) |
---|---|

Pages (from-to) | 147-158 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2912 |

State | Published - 2004 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

**Fixed-Location Circular-Arc Drawing of Planar Graphs.** / Efrat, Alon; Erten, Cesim; Kobourov, Stephen G.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Fixed-Location Circular-Arc Drawing of Planar Graphs

AU - Efrat, Alon

AU - Erten, Cesim

AU - Kobourov, Stephen G

PY - 2004

Y1 - 2004

N2 - In this paper we consider the problem of drawing a planar graph using circular-arcs as edges, given a one-to-one mapping between the vertices of the graph and a set of n points on the plane, where n is the number of vertices in the graph. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n 2) algorithm to find out if a drawing with no crossings can be realized. We present an improved O(n 7/4 polylog n) time algorithm. For the special case where the possible circular arcs for each edge are of the same length, we present an even more efficient algorithm that runs in O(n 3/2 polylog n) time. We also consider the problem if we have more than two possible circular arcs per edge and show that the problem becomes NP-Hard. Moreover, we show that two optimization versions of the problem are also NP-Hard.

AB - In this paper we consider the problem of drawing a planar graph using circular-arcs as edges, given a one-to-one mapping between the vertices of the graph and a set of n points on the plane, where n is the number of vertices in the graph. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n 2) algorithm to find out if a drawing with no crossings can be realized. We present an improved O(n 7/4 polylog n) time algorithm. For the special case where the possible circular arcs for each edge are of the same length, we present an even more efficient algorithm that runs in O(n 3/2 polylog n) time. We also consider the problem if we have more than two possible circular arcs per edge and show that the problem becomes NP-Hard. Moreover, we show that two optimization versions of the problem are also NP-Hard.

UR - http://www.scopus.com/inward/record.url?scp=35048817393&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35048817393&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:35048817393

VL - 2912

SP - 147

EP - 158

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -