Drawing Planar Graphs with Few Geometric Primitives

Abstract

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n − 17)/3 segments on an O(n) × O(n 2 ) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/2 edges on an O(n) × O(n 2 ) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n − 11)/3 arcs. This is significantly smaller than the lower bound of 2n for line segments for a nontrivial graph class. * A preliminary version of this work appeared in: Proc. 43rd Int. Workshop Graph-Theor. Concepts Comput. Sci. (WG'17) [14]. The work of P. Kindermann and A.

Drawing Planar Graphs with Few Segments on a Polynomial Grid

“…Our contribution is summarized in Table 1. In Section 2, we show that every tree has a drawing with at most 3n/4 − 1 segments on the n × n grid, improving the area bound by Hültenschmidt et al [13]. We then focus on drawing 3-connected planar graphs in Section 3.…”

“…Dujmović et al [7] were the first to study drawings with few segments and provided upper and lower bounds for several planar graph classes. Since then, several new results have been provided ( [8,13,14,20,21], refer also to Table 1). These results shed only a little light on the area requirements of the drawings.…”

Experimental Analysis of the Accessibility of Drawings with Few Segments

“…Finally, the algorithm FewSegments is based on the algorithm by Hülten-schmidt et al [6] that draws trees on a quasi-polynomial grid with a minimum number of segments. On a high level, that algorithm uses a heavy path decomposition of a tree, which decomposes the tree in heavy edges and light edges.…”

“…When the algorithm assigns a vector to a subtree, we allow it to increase the length of the vector slightly such that the new vector is an integer multiple of a smaller primitive vector. For example, if the algorithm would assign a vector (6,11), then this heuristic would change the vector to (6,12). This implies that the segments on the heavy path in this subtree do not have to use vectors that are integer multiples of (6, 11), but only integer multiples of (1,2).…”

“…It is unknown however, if every tree can be drawn with n odd /2 segments using only a polynomial grid size. We use a heuristic based on the algorithm of Hültenschmidt et al [6] that draws a tree with minimal visual complexity and quasi-polynomial area and compare its drawing in the user study against drawings of other algorithms. In particular, we use the algorithms of Walker II [14] and of Rusu et al [11] as alternatives.…”