La Innegociabilidad del Ejercicio de las Potestades Públicas

Orthogonal Drawings. In an orthogonal drawing Γ of a graph G = (V; E), each vertex is represented as a point in the plane whereas each edge e is represented

as a sequence of horizontal and vertical line segments Γ(e). In addition, each vertex v is associated with four distinct ports called north, west, south and east. If the segment of the representation of edge e that is attached to v has v as its bottom (right, top, left) endpoint, we say that e uses v ’s north (west, south, east; respectively) port. As a further restriction to orthogonal drawings, each of the ports of a vertex can be used only by one edge. Hence, the maximum degree of a graph admitting an orthogonal drawing is at most four. For an example of an orthogonal drawing refer to Fig. 2.3b.

Orthogonal graph drawing dates back to applications in VLSI design and floor- planning which where studied in the early 1980’s [125, 127, 156]. Nowadays, the most important applications include UML or BPMN diagrams. Hence, it comes at no surprise that the orthogonal graph drawing model has been deeply studied. For an overview of well-established techniques we refer the reader to [63, 119, 150] while we point the reader to [43, 95, 129, 149] for some efficient algorithmic solutions providing high quality drawings.

By definition, orthogonal drawings achieve perfect angular resolution for graphs of maximum degree four while they also avoid acute angles at the bends of edges. These properties contribute to the good readability of orthogonal drawings. Addi- tionally, orthogonal graph drawing algorithms usually also try to optimize the area and the curve complexity of the resulting layout. For the sake of brevity, we will call an orthogonal drawing of curve complexity k an OCk-drawing . It is known that

all planar graphs of maximum degree four admit a planar OC3-drawing with the

exception of the octahedral graph which admits a planar OC4-drawing [43, 129];

these drawings require O(n) × O(n) area. Moreover, if the maximum degree of a planar graph is three, it admits a planar OC2-drawing [117].

It is also known that each connected graph of maximum degree four admits a (not necessarily planar) OC3-drawing with no guarantees regarding the number of

intersections per edge [43].

The Topology-Shape-Metrics framework. Finding an orthogonal drawing with a minimum number of bends over all embeddings of a maximum degree four planar graph has been shown to be NP -hard [100]. For plane graphs of maxi- mum degree four, it can however be solved in polynomial time by reduction to two min-cost flow problems [149]. More precisely, given the embedding (that is, a topology ), the solution of the first network flow defines an orthogonal representa- tion, which defines the angles between any two edges incident to the same vertex and the bends encountered on each edge. The second network flow problem is based on a feasible representation (or shape) and the flow algorithm computes how long each edge should be (that is, the metrics of the drawing) in order to find a drawing that realizes the given representation. We say that an orthogonal drawing Γ realizes an orthogonal representation R, if the angles between any pair of edges incident to the same vertex in Γ is identical to the angle prescribed in R

(a) (b) (c) (d) (e)

Figure 2.3: Five drawings of the same graph using different drawing styles: (a) Straight-line drawing, (b) Orthogonal drawing, (c) Smooth orthogonal drawing, (d) Octilinear drawing, and, (e) Kandinsky drawing.

and if in Γ, the bends occuring on each edge are the ones described by R. This framework of graph drawing is also known as the topology-shape-metrics approach as the drawing is produced in three steps (where the first topology step is to decide for an embedding). It is worth pointing out that each feasible orthogonal repre- sentation also can be realized. This makes it so that each of the algorithms used for solving one of the three steps of the framework are interchangeable without influencing the other steps.

In the following, we discuss two extensions of the orthogonal graph drawing model which each allow one more type of geometric primitives in the representa- tions of edges, namely smooth orthogonal drawings [31] which also allow circular arcs and octilinear drawings which also allow diagonal straight-line segments. Af- terwards, we discuss Kandinsky drawings [95] which generalize orthogonal drawings to graphs of larger maximum degree.

Smooth Orthogonal Drawings. A smooth orthogonal drawing Γ of a graph

G = (V; E) maps each vertex v ∈ V to a point Γ(v ) ∈ R2 and each edge e ∈ E to a sequence of straight-line segments and quarter, half and three-quarter circular arcs such that the tangents of consecutive segments overlap at their shared bend. In addition, the port restriction of the orthogonal model is also used in smooth orthogonal drawings. As in orthogonal drawings, smooth orthogonal drawings can only visualize graphs of maximum degree at most four. For an example of a smooth orthogonal drawing refer to Fig. 2.3c. Observe that the orthogonal drawing of the same graph in Fig. 2.3b has curve complexity three which is in fact necessary while there even exists a (less symmetric) smooth orthogonal drawing of curve complexity one.

Smooth orthogonal drawings have been proposed by Bekos et al. [31] as a drawing style which combines the easy readability and rigidity of the well established orthogonal drawing style with the artistic appeal and smoothness of Lombardi drawings2 _{[82, 86]. As a result, smooth orthogonal drawings achieve optimal}

2_{In a Lombardi drawing, edges are represented by circular arc segments. In addition, all angles}

angular resolution for graphs of maximum degree four while also realizing each edge with a smooth curve. Similar to the orthogonal graph drawing model, other important parameters to evaluate the quality of smooth orthogonal drawings are the area and the curve complexity of the layout. We will refer to a smooth orthogonal drawing of curve complexity k as an SCk drawing .

It is known that planar SC1-drawings always exist for maximum degree three

planar graphs [29] and for outerplanar graphs of maximum degree four [9] both requiring superpolynomial area. In contrast, there exist maximum degree four planar graphs which do not admit a planar SC1-drawing [31]. On the other hand,

all planar graphs of maximum degree four admit a planar SC2-drawing. The

required area may however be exponential for graphs of maximum degree four, while it is cubic for graphs of maximum degree three [9]. The problem of deciding if a planar graph of maximum degree four admits a planar SC1-drawing is open

and has been conjectured to be NP -hard [9].

In addition, each graph of maximum degree four admits an SC1-drawing with

no guarantees regarding the number of intersections per edge [29].

Octilinear Drawings. In an octilinear drawing , each vertex is represented by a point in the plane whereas each edge is drawn as a sequence of straight-line segments at slopes {0; −1; 1; ∞}. In contrast to orthogonal drawings and smooth orthogonal drawings, each vertex is assigned eight ports (north, north-west, west, south-west, south, south-east, east, and north-east) and as a consequence octilin- ear drawings can realize graphs of maximum degree at most eight. More intuitively speaking, octilinear drawings add two additional slopes for segments to the orthog- onal graph drawing model. For an example octilinear drawing refer to Fig. 2.3d. Observe that the curve complexity of the example drawing is two whereas the orthogonal drawing of the same graph in Fig. 2.3b has curve complexity three. Moreover, when comparing the octilinear layout to the smooth orthogonal one in Fig. 2.3c, we can observe that in both drawings all bends and all vertices have the same position. We will investigate this observation further in Section 3.1.

The most common applications of octilinear drawings include metro-maps and map schematization [110, 135, 134, 148]. In addition, octilinear drawings gen- eralize the concept of orthogonal drawings for graphs with maximum degree at most eight while also a variant called slanted orthogonal drawings3 _{has been in-}

vestigated in the literature [32, 35] for representing nonplanar graphs of maximum degree four. The most important aesthetic criteria for octilinear drawings include area, curve complexity and smoothness of edges since diagonal and axis-aligned segments can meet at their shared bend either at ı₄ or 3ı₄ . As with orthogonal and smooth orthogonal drawings we will refer to an octilinear drawing of curve complexity k as an 8Ck drawing .

3_{In slanted orthogonal drawings, vertices are incident to axis-aligned segments while intersec-}

It has been shown that all planar graphs of maximum degree eight admit a planar 8C3-drawing [121]. Moreover, for planar graphs of maximum degree five it

is possible to compute a planar 8C2-drawing which even fits in polynomial area if

the maximum degree is at most four [28]. In addition, planar graphs of maximum degree three always admit a planar 8C1-drawing [69, 116]. The problem of deciding

whether a planar graph of maximum degree eight admits a planar 8C1-drawing has

been shown to be NP -hard [135] while it is not known if NP -hardness is still true for planar graphs of lower maximum degree. In particular, this question is especially interesting for planar graphs of maximum degree four since as mentioned earlier planar graphs of maximum degree three are always positive instances whereas there exist planar graphs of maximum degree four without a planar 8C1-drawing [34].

Kandinsky Drawings. In a Kandinsky drawing Γ of a graph G = (V; E) [41, 62, 95, 96], each vertex v ∈ V is represented by an axis-aligned square Γ(v ) and each edge (u; v ) ∈ E is represented by a sequence of axis-aligned straight-line segments Γ((u; v )) whose endpoints touch Γ(u) and Γ(v ) at exactly one point. In the classic model, the squares representing vertices have uniform side length ∆ where ∆ denotes the maximum degree of G so that the side to which each edge attaches can be chosen arbitrarily. Moreover, in the case of grid drawings, vertices are centered on a coarse grid whose grid lines are sufficiently far apart to avoid vertices to overlap (i.e. at Ω(∆) distance) whereas bends of edges are located on a fine grid (usually fine grid lines have unit distance). For an example of a Kandinsky drawing refer to Fig. 2.3e.

Kandinsky drawings relax the strict port constraints of the orthogonal graph drawing style to allow drawings for graphs of arbitrary degree while also it may be possible to reduce the curve complexity for graphs of maximum degree four (compare for instance the drawings in Figs. 2.3b and 2.3e). They find wide usage in VLSI design and UML and BPMN diagrams where usually vertices are realized by two-dimensional objects due to application constraints. Due to their practical applications, Kandinsky drawings have received much attention in research over the decades, see for instance [36, 44, 45].

The Kandinsky drawing model allows for natural extensions to smooth orthog- onal and octilinear drawings which so far have only received little attention in the literature. A topological book embedding (as defined in Section 2.3.3) can be used to produce a planar SC2-Kandinsky drawing for any planar graph in linear time [31].

The resulting drawings have quadratic area and all vertices are located on a line while edges are represented by half circular arcs or as a sequence of two half cir- cular arcs (one above and one below the line). Further, it was shown that not all planar graphs admit an SC1-Kandinsky drawing [29]. In the octilinear setting

the most notable application of the Kandinsky style are the so-called Sloginsky drawings [33, 36] which generalize slanted orthogonal drawings to higher maxi- mum degree. As in slanted orthogonal drawings, the additional slopes are used for

segments on which intersections occur to make them easier distinguishable from vertices.