Strongly Monotone Drawings of Planar Graphs

A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices. We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs are based on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex

Non-Homotopic Drawings of Multigraphs

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$, which is a big improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is $\Omega\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.Comment: 19 page

Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $\Gamma$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+\epsilon$ allows us to draw every graph. Namely, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+\epsilon$. Third, our drawings with spanning ratio smaller than $1+\epsilon$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio

On generalized strongly monotone drawings of planar graphs (Recent Trends in Algorithms and Computation)

グラフの描画方法には大きな任意性があり, さまざまな観点から見やすいグラフの描画方法が研究されている. 本稿では, パスを見つけやすいグラフの描画として提案されている強単調性描画に若目し, それを精密化した概念として, a-強単調性描画を提案する. 木, 極大平面的グラフ, 平面的3—木, 2-連結外平面的グラフについて, 既存研究で強単調性描画可能であることがわかっているが, 本稿では, それらについてa—強単調性描画可能なaの値の評価を行い, それらの結果の精密化を行う