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

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

On Planar Greedy Drawings of 3-Connected Planar Graphs

A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra\u27s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture

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

Complete combinatorial characterization of greedy-drawable trees

A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices $s,t$ ($s \neq t$), there is a neighbor vertex of $s$ that is closer to $t$ than to $s$ in the Euclidean distance. Greedy drawings are important in the context of message routing in networks, and graph classes that admit greedy drawings have been actively studied. N\"{o}llenburg and Prutkin (Discrete Comput. Geom., 58(3), pp.543-579, 2017) gave a characterization of greedy-drawable trees in terms of an inequality system that contains a non-linear equation. Using the characterization, they gave a linear-time recognition algorithm for greedy-drawable trees of maximum degree $\leq 4$. However, a combinatorial characterization of greedy-drawable trees of maximum degree 5 was left open. In this paper, we give a combinatorial characterization of greedy-drawable trees of maximum degree $5$, which leads to a complete combinatorial characterization of greedy-drawable trees. Furthermore, we give a characterization of greedy-drawable pseudo-trees.Comment: 26 pages, 30 fugure

Monotone Simultaneous Paths Embeddings in Rd\mathbb{R}^d

International audienceWe study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding