Higher-Degree Orthogonal Graph Drawing with Flexibility Constraints

Much work on orthogonal graph drawing has focused on 4-planar graphs, that is planar graphs where all vertices have maximum degree 4. In this work, we study aspects of the Kandinsky model, which is a model for orthogonal graph drawings of higher-degree graphs. First, we examine the decision problem β-Embeddability, which asks whether for a given planar graph with a fixed or variable embedding, a drawing in the Kandinsky model exists where every edge has at most β bends. We show that 0-Embeddability in the Kandinsky model is equivalent to 0-Embeddability in the lower-degree case. We show that 1-Embeddability for multigraphs with variable planar embeddings is NP-complete. Then, we show that any simple graph is 1-embeddable, even if it has a fixed planar embedding, and we present a linear-time algorithm for finding a corresponding 1-bend drawing. Furthermore, we show to find a 2-bend Kandinsky drawing of any plane graph in linear time. Next, we study some restrictions of the bend minimization problem OptimalKandinskyDraw, which finds a Kandinsky drawing with the minimum number of bends for a plane graph. We present a lineartime algorithm solving OptimalKandinskyDraw for biconnected, outerplanar, inner-triangulated graphs. Then, we give an O(n3) time algorithm for finding bend-minimal 1-bend Kandinsky drawings of series-parallel graphs. Finally, we inquire into new ways to solve OptimalKandinskyDraw using linear programming. We show that for any constant c ∈ R a graph exists so that the difference between the real solution and the integer solution of the corresponding linear program is greater than c.