Monotone Drawings of kk-Inner Planar Graphs

A $k$-inner planar graph is a planar graph that has a plane drawing with at most $k$ {internal vertices}, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a $0$-inner planar graph. In this paper, we show how to construct a monotone drawing of a $k$-inner planar graph on a $2(k+1)n \times 2(k+1)n$ grid. In the special case of an outerplanar graph, we can produce a planar monotone drawing on a $n \times n$ grid, improving previously known results.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018). Revised introductio

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