On the Total Number of Bends for Planar Octilinear Drawings

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45◦ line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. As the problem of finding planar octilinear drawings of minimum number of bends is NP-hard [16], in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. [14] on the slope number of planar graphs, we can derive an upper bound of 4n − 10 bends for 8-planar graphs with n vertices. We considerably improve this general bound and corresponding previous ones for triconnected 4-, 5and 6-planar graphs. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia [19]. 1 Motivation and Background Octilinear drawings of graphs have a long history of research, which dates back to the early thirteenth century, when an English technical draftsman, Henry Charles Beck (also known as Harry Beck), designed the first schematic map of London Underground. His map, the so-called Tube map, looked more like an electrical circuit diagram (consisting of horizontal, vertical and diagonal line segments) rather than a true map, as the underlying geographic accuracy was neglected. Laying out networks in such a way is called octilinear graph drawing and plays an important role in map-schematization and the design of metro-maps. In particular, an octilinear drawing Γ(G) of a graph G = (V,E) is one in which each vertex occupies a point on an integer grid and each edge is drawn as a sequence of horizontal, vertical and diagonal at 45◦ line segments. When G is planar, usually it is required Γ(G) to be planar as well. In planar octilinear graph drawing, an important goal is to keep the number of bends small, so that the produced drawings can be understood easily. However, the problem of determining whether a given embedded planar graph of maximum degree eight admits a bend-less planar octilinear drawing is NP-complete [16]. This motivated us to neglect optimality and study upper and lower bounds on the total number of bends of such drawings. Surprisingly enough, very few results were known, even if the octilinear model has been extensively studied in the areas of metro-map visualization and map schematization. One can derive the first (non-trivial) upper bound on the required number of bends from a result on the planar slope number of graphs by Keszegh et al. [14], who proved that every k-planar graph (that is, planar of maximum degree k) has a planar drawing with at most d2e different slopes in which each edge has at most two bends. For 3 ≤ k ≤ 8, the drawings are octilinear, which yields an upper bound of 6n − 12, where n is the number of vertices of the graph. The bound can be reduced to 4n− 10 with some effort; see our subsection on related work. On the other hand, it is known that every 3-planar graph with five or more vertices admits a planar octilinear drawing in which all edges are bend-less [13, 7]. Also, for 4 ≤ k ≤ 5, it was 1 ar X iv :1 51 2. 04 86 6v 1 [ cs .C G ] 1 5 D ec 2 01 5 Table 1: A short summary of our results. Upper bounds Graph class Lower bound Ref. Previous Ref. New Ref. 3-con. 4-planar n/3− 1 Thm. 4 2n [2] n+ 5 Thm. 1 3-con. 5-planar 2n/3− 2 Thm. 4 5n/2 [2] 2n− 2 Thm. 2 3-con. 6-planar 4n/3− 6 Thm. 4 4n− 10 [14] 3n− 8 Thm. 3 recently proved that 4and 5-planar graphs admit planar octilinear drawings with at most one bend per edge [2], which implies that the total number of bends for 4and 5-planar graphs can be upper bounded by 2n and 5n/2, respectively. The remainder of this paper is organized as follows. In Section 2, we considerably improve all aforementioned bounds for the classes of triconnected 4-, 5and 6-planar graphs. In Section 3, we present corresponding lower bounds for these three classes of planar graphs. We conclude in Section 4 with open problems and future work. For a summary of our results also refer to Table 1. 1.1 Related work. As already stated, Keszegh et al. [14] have proved that every k-planar graph admits a planar drawing with at most d2e different slopes in which each edge has at most two bends. If one appropriately adjusts the slopes of all edge segments incident to a vertex, then one can show that any k-planar graph, with 3 ≤ k ≤ 8, admits a planar octilinear drawing in which each edge has at most two bends. This implies that any k-planar graph on n vertices can have at most 6n − 12 bends, where 3 ≤ k ≤ 8. One can easily improve this bound to 4n − 10 as follows. The edge that “enters” a vertex from its south port and the edge that “leaves” each vertex from its top port in the s-t ordering of the algorithm of Keszegh et al. can both be drawn with only one bend each. Since each vertex is incident to exactly two such edges (except for the first and last ones in the s-t ordering which are only incident to one such edge each), it follows that 2n − 2 edges can be drawn with at most one bend. Hence, the bound of 4n− 10 follows. Octilinear drawings form a natural extension of the so-called orthogonal drawings, which allow for horizontal and vertical edge segments only. For such drawings, the bend minimization problem can be solved efficiently, assuming that the input is an embedded graph [19]. However, the corresponding minimization problem over all embeddings of the input graph is NP-hard [10]. Note that in [19] the author describes how one can extend his approach, so to compute a bend-optimal octilinear representation1 of any given embedded 8-planar graph. However, such a representation may not be realizable by a corresponding planar octilinear drawing [5]. For orthogonal drawings, several bounds on the total number of bends are known. Biedl [3] presents lower bounds for graphs of maximum degree 4 based on their connectivity (simply connected, biconnected or triconnected), planarity (planar or not) and simplicity (simple or non-simple with multiedges or selfloops). It is also known that any 4-planar graph (except for the octahedron graph) admits a planar orthogonal drawing with at most two bends per edge [4, 15]. Trivially, this yields an upper bound of 4n bends, which can be improved to 2n+ 2 [4]. Note that the best known lower bound is due to Tamassia et al. [20], who presented 4-planar graphs requiring 2n− 2 bends. Finally, in metro-map visualization many approaches have been proposed that result in octilinear or nearly-octilinear drawings (see, e.g., [11, 16, 17, 18]). However, most of them are heuristics and therefore do not focus on the bend-minimization problem explicitly. Recall that a representation of a graph describes the angles and the bends of a drawing, neglecting its exact geometry [19].