Planar and Poly-arc Lombardi Drawings

In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.

[1]  Achilleas Papakostas,et al.  On the Angular Resolution of Planar Graphs , 1994, SIAM J. Discret. Math..

[2]  Michael T. Goodrich,et al.  A Framework for Drawing Planar Graphs with Curves and Polylines , 1998, J. Algorithms.

[3]  J. S. Andrade,et al.  Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. , 2004, Physical review letters.

[4]  Roberto Tamassia,et al.  Curvilinear Graph Drawing Using the Force-Directed Method , 2004, GD.

[5]  David Eppstein,et al.  Optimal 3D Angular Resolution for Low-Degree Graphs , 2013, J. Graph Algorithms Appl..

[6]  Giuseppe Di Battista,et al.  Angles of Planar Triangular Graphs , 1996, SIAM J. Discret. Math..

[7]  Nicolaos Matsakis.  Transforming a random graph drawing into a Lombardi drawing , 2010, ArXiv.

[8]  Jarke J. van Wijk,et al.  Force‐Directed Edge Bundling for Graph Visualization , 2009, Comput. Graph. Forum.

[9]  David Eppstein,et al.  Drawing Trees with Perfect Angular Resolution and Polynomial Area , 2010, Graph Drawing.

[10]  Ulrik Brandes,et al.  Using Graph Layout to Visualize Train Interconnection Data , 1998, Graph Drawing.

[11]  David Eppstein,et al.  Delta-Confluent Drawings , 2005, Graph Drawing.

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

[13]  Petra Mutzel,et al.  Planar Polyline Drawings with Good Angular Resolution , 1998, GD.

[14]  Michael Hirsch,et al.  Biclique Edge Cover Graphs and Confluent Drawings , 2006, Graph Drawing.

[15]  Michael T. Goodrich,et al.  Force-Directed Lombardi-Style Graph Drawing , 2011, Graph Drawing.

[16]  David Eppstein,et al.  Confluent Layered Drawings , 2006, Algorithmica.

[17]  José S. Andrade,et al.  Erratum: Apollonian Networks: Simultaneously Scale-Free, Small World, Euclidean, Space Filling, and with Matching Graphs [Phys. Rev. Lett. 94 , 018702 (2005)] , 2009 .

[18]  Michael T. Goodrich,et al.  Drawing planar graphs with circular arcs , 2001, Discret. Comput. Geom..

[19]  Roberto Tamassia,et al.  Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract) , 1994, ESA.