Monotone drawings of graphs with few directions

https://doi.org/10.1016/j.ipl.2016.12.004Get rights and content

Highlights

  • We define k-monotone drawings, where paths are monotone in one of k directions.

  • We prove that a graph admits a 1-monotone drawing if and only if it can be made Hamiltonian by adding at most one edge.

  • We prove that maximal planar graphs admit 2-monotone drawings.

  • We prove that triconnected planar graphs admit 3-monotone drawings.

Abstract

A monotone drawing of a graph G is a straight-line planar drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Thus, a path towards every possible destination vertex can be easily found in a monotone drawing by following a certain direction. However, since different pairs of vertices may use different directions of monotonicity, selecting the correct direction to follow may become as complicated as the original task of finding the path. To overcome this limitation, in this paper we study the existence of 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 has a Hamiltonian path between two vertices lying in the same face. 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 based on Schnyder realizers.

Introduction

A recent trend in the research on the visualization of graphs is the one of constructing drawings in which paths follow some kind of geodesic curve, as this feature has been proved crucial to improve readability [1], especially with respect to the ability of the user to perform some path-finding tasks. Examples of this type of drawings include greedy drawings [2], [3], [4], [5], self-approaching [6] and increasing-chord drawings [7], [8], and monotone drawings [9], which are the main subject of this paper.

A monotone drawing of a graph is a planar straight-line drawing in which every pair of vertices is connected by a path that is monotone with respect to some direction. In the original paper [9] on this topic it was proved that trees and biconnected planar graphs always admit monotone drawings, and this is true even if the embedding of the graph is prescribed in advance [10]. The algorithm for trees has been later improved to obtain drawings with asymptotically-optimal area [11]. On the other hand, not all planar graphs admit such a drawing with a given embedding [9], but it was recently proved that they do if a suitable embedding is chosen [12].

The original motivation [9] for the introduction of monotone drawings was to help a reader in finding a path in the graph by following some direction, starting from the origin vertex, while seeking for the target vertex. The explanatory example provided in that paper, indeed, was based on a typical situation in which a tourist rotates her city-map in order to better understand the way to the destination. However, the advantages of this type of feature start to be perceptible only if the direction to follow is easy to determine; in other words, the tourist should be able to easily select a rotation of the map that satisfies her needs.

In this respect, the notion of strong monotonicity is arguably the most suited and natural for the original purpose. In a strongly monotone drawing [9], in fact, the direction of monotonicity for a path between a source and a destination is required to be parallel to the line through them. On the other hand, this requirement seems to be too restricted, as strongly monotone drawings have been proved to always exist only for trees [13], for planar 3-trees [8], and only recently for larger graph classes [14], including 3-connected planar graphs. Exponential area has been proved to be necessary for some trees in this drawing style [15].

In this paper we hence propose another variant of monotone drawings, which is still aimed at simplifying the selection of the direction to follow, and hence at improving their practical applicability. Namely, we want to construct monotone drawings with the additional requirement that the total number of different directions of monotonicity is bounded by some (possibly constant) function. In the city-map metaphor, this would bound the number of rotations of the map that a tourist has to perform in order to find a direction that satisfies her needs. More formally, we define k-monotone drawings, that is, monotone drawings for which there exists a set of k directions such that every pair of vertices is connected by a path that is monotone with respect to one of these directions.

We study 1-monotone drawings in Section 3, and give a characterization of these graphs in terms of the existence of a specific type of Hamiltonian path. Then, in Section 4 we prove that all maximal planar graphs admit a 2-monotone drawing, which is a tight bound since not all such graphs have a Hamiltonian path, and we extend the technique to triconnected planar graphs, constructing 3-monotone drawings for them. Both these results are obtained by exploiting properties of the well-known Schnyder drawings [16], [17]. We give some preliminaries and definitions in Section 2 and some conclusive remarks in Section 5, where we also list some open problems.

Section snippets

Definitions and preliminaries

A planar drawing of a graph is a mapping of each vertex to a distinct point of the plane and of each edge to a curve connecting its endpoints such that edges do not cross except, possibly, at common endpoints. A graph is planar if it admits a planar drawing. A planar drawing partitions the plane into connected regions, called faces. The unbounded face is the outer face. A straight-line planar drawing is a planar drawing in which all edges are represented by straight-line segments.

A graph is

1-Monotone drawings of planar graphs

In this section we characterize the graphs admitting a 1-monotone drawing as the planar graphs that either are Hamiltonian or can be made so by adding at most one edge, while maintaining planarity.

We start by describing an algorithm to construct a 1-monotone drawing of any graph satisfying this condition.

Lemma 1

Let G=(V,E) be a graph with a Hamiltonian path P(s,t) between two vertices s and t such that graph G=(V,E{(s,t)}) is planar. Then, G admits a 1-monotone drawing.

Proof

The idea is to construct a

k-Monotone drawings of triconnected planar graphs

In this section we study the existence of k-monotone drawings of triconnected planar graphs with small k. Clearly, not all these graphs admit a drawing with k=1, by Theorem 1, as there exist triconnected planar graphs not containing any Hamiltonian path, and this is true even if we restrict to maximal planar graphs.

We prove that maximal planar graphs can always be realized with k=2, which is hence a tight bound, while for general triconnected planar graphs we can only prove it for k=3. Both

Conclusions and open problems

In this paper we studied the possibility of constructing monotone drawings of planar graphs that use as few directions of monotonicity as possible, at the aim of allowing a user to easily find the direction to follow when navigating a drawing from a vertex to another.

We characterized the graphs that can be realized with a single direction in terms of the existence of a Hamiltonian path whose endvertices can be made incident to the same face in a planar drawing of the graph. Also, we proved that

Acknowledgements

The author would like to gratefully thank Giuseppe Di Battista for useful discussions. The research on this work was partially supported by DFG grant Ka812/17-1. A preliminary version of this work appeared in [29].

References (29)

  • M. Nöllenburg et al.

    On self-approaching and increasing-chord drawings of 3-connected planar graphs

  • P. Angelini et al.

    Monotone drawings of graphs

    J. Graph Algorithms Appl.

    (2012)
  • P. Angelini et al.

    Monotone drawings of graphs with fixed embedding

    Algorithmica

    (2015)
  • D. He et al.

    Optimal monotone drawings of trees

  • Cited by (0)

    View full text