### Abstract

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.

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

Pages (from-to) | 3-22 |

Number of pages | 20 |

Journal | Algorithmica |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2013 |

### Fingerprint

### Keywords

- Cartogram
- Contact representation
- Graph drawing
- Planar graph
- Polygon

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*,

*67*(1), 3-22. https://doi.org/10.1007/s00453-013-9764-5

**Linear-time algorithms for hole-free rectilinear proportional contact graph representations.** / Alam, M. Jawaherul; Biedl, Therese; Felsner, Stefan; Gerasch, Andreas; Kaufmann, Michael; Kobourov, Stephen G.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 67, no. 1, pp. 3-22. https://doi.org/10.1007/s00453-013-9764-5

}

TY - JOUR

T1 - Linear-time algorithms for hole-free rectilinear proportional contact graph representations

AU - Alam, M. Jawaherul

AU - Biedl, Therese

AU - Felsner, Stefan

AU - Gerasch, Andreas

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

PY - 2013/9

Y1 - 2013/9

N2 - In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.

AB - In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.

KW - Cartogram

KW - Contact representation

KW - Graph drawing

KW - Planar graph

KW - Polygon

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UR - http://www.scopus.com/inward/citedby.url?scp=84880231167&partnerID=8YFLogxK

U2 - 10.1007/s00453-013-9764-5

DO - 10.1007/s00453-013-9764-5

M3 - Article

AN - SCOPUS:84880231167

VL - 67

SP - 3

EP - 22

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -