Monotone drawings of graphs with few directions

A monotone drawing of a graph is a straight-line planar drawing such that every pair of vertices is connected by a path that monotonically increases with respect to a direction. However, different pairs of vertices might use different directions of monotonicity. In this paper we aim at constructing monotone drawings in which the number of different directions of monotonicity used by all the pairs of vertices is small. We show that a planar graph admits a monotone drawing with only one direction of monotonicity if and only if it is Hamiltonian. Also, we prove that maximal planar graphs admit monotone drawings with two (orthogonal) directions, while triconnected planar graphs with three directions. The latter two results are obtained by applying the famous drawing algorithm by Schnyder.

[1]  Md. Iqbal Hossain,et al.  Monotone Grid Drawings of Planar Graphs , 2014, FAW.

[2]  Stefan Felsner,et al.  Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes , 2001, Order.

[3]  Paul C. Kainen,et al.  The book thickness of a graph , 1979, J. Comb. Theory, Ser. B.

[4]  Antonios Symvonis,et al.  Monotone Drawings of Graphs with Fixed Embedding , 2013, Algorithmica.

[5]  Christos H. Papadimitriou,et al.  On a conjecture related to geometric routing , 2004, Theor. Comput. Sci..

[6]  Roberto Tamassia,et al.  Algorithms for Plane Representations of Acyclic Digraphs , 1988, Theor. Comput. Sci..

[7]  Joachim Gudmundsson,et al.  Increasing-Chord Graphs On Point Sets , 2014, Graph Drawing.

[8]  Luca Grilli,et al.  An Algorithm to Construct Greedy Drawings of Triangulations , 2010, J. Graph Algorithms Appl..

[9]  Emilio Di Giacomo,et al.  Lower and Upper Bounds for Long Induced Paths in 3-Connected Planar Graphs , 2013, WG.

[10]  Martin Nöllenburg,et al.  Euclidean Greedy Drawings of Trees , 2013, ESA.

[11]  Frank Thomson Leighton,et al.  Some Results on Greedy Embeddings in Metric Spaces , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[12]  Alexander Wolff,et al.  On Monotone Drawings of Trees , 2014, Graph Drawing.

[13]  Esther M. Arkin,et al.  On monotone paths among obstacles with applications to planning assemblies , 1989, SCG '89.

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

[15]  Martin Nöllenburg,et al.  On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs , 2014, Graph Drawing.

[16]  Timothy M. Chan,et al.  Self-approaching Graphs , 2012, Graph Drawing.

[17]  Roberto Tamassia,et al.  Output-Sensitive Reporting of Disjoint Paths , 1996, Algorithmica.

[18]  Günter Rote Strictly convex drawings of planar graphs , 2005, SODA.

[19]  Weidong Huang,et al.  A graph reading behavior: Geodesic-path tendency , 2009, 2009 IEEE Pacific Visualization Symposium.