Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

[1]  Michael T. Goodrich,et al.  Planar Drawings of Higher-Genus Graphs , 2009, J. Graph Algorithms Appl..

[2]  Raghavan Dhandapani,et al.  Greedy Drawings of Triangulations , 2008, SODA '08.

[3]  Hiroshi Nagamochi,et al.  Convex drawings of graphs with non-convex boundary constraints , 2006, Discret. Appl. Math..

[4]  Anne Verroust-Blondet,et al.  Computing a canonical polygonal schema of an orientable triangulated surface , 2001, SCG '01.

[5]  Michael T. Goodrich,et al.  A multi-dimensional approach to force-directed layouts of large graphs , 2000, Comput. Geom..

[6]  M. Chrobak,et al.  Convex Grid Drawings of 3-Connected Planar Graphs , 1997, Int. J. Comput. Geom. Appl..

[7]  David Harel,et al.  Drawing graphs nicely using simulated annealing , 1996, TOGS.

[8]  Hiroshi Nagamochi,et al.  Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs , 1996, Algorithmica.

[9]  Goos Kant,et al.  Drawing planar graphs using the canonical ordering , 1996, Algorithmica.

[10]  Marek Chrobak,et al.  Convex drawings of graphs in two and three dimensions (preliminary version) , 1996, SCG '96.

[11]  Kozo Sugiyama,et al.  Graph Drawing by the Magnetic Spring Model , 1995, J. Vis. Lang. Comput..

[12]  Marek Chrobak,et al.  A Linear-Time Algorithm for Drawing a Planar Graph on a Grid , 1995, Inf. Process. Lett..

[13]  Edward M. Reingold,et al.  Graph drawing by force‐directed placement , 1991, Softw. Pract. Exp..

[14]  János Pach,et al.  How to draw a planar graph on a grid , 1990, Comb..

[15]  Bernd Becker,et al.  On the Optimal Layout of Planar Graphs with Fixed Boundary , 1987, SIAM J. Comput..

[16]  J. Geelen.  ON HOW TO DRAW A GRAPH , 2012 .

[17]  B Imre,et al.  Strictly Convex Drawings of Planar Graphs , 2006 .

[18]  I. Tollis,et al.  Graph Drawing , 1994, Lecture Notes in Computer Science.

[19]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[20]  W. T. Tutte.  Convex Representations of Graphs , 1960 .

[21]  K. Wagner.  Bemerkungen zum Vierfarbenproblem. , 1936 .

[22]  Ôôöøøøøóò Óó,et al.  Strictly Convex Drawings of Planar Graphs , 2022 .