Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Abstract

Abstract. The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.

Untangling a Planar Graph

“…We further prove that MINSHIFTEDVERTICES is hard to approximate in the following sense: if there is a real ε ∈ (0, 1] and a polynomial-time algorithm that guarantees to untangle any drawing δ of any n-vertex planar graph G with at most (n 1−ε ) · (shift(G, δ) + 1) moves, then P = N P (Theorem 3.3). • We complement the complexity result of Kaufmann and Wiese [11] on 1BEND-POINTSETEMBEDDABILITY by showing that it is NP-hard to decide whether a given one-to-one correspondence between the vertices of a planar graph G and a planar point set S extends into a plane drawing of G with at most one bend per edge (Theorem 3.4). We also show that the problem lies in PSPACE (Theorem 3.6) and that an optimization version of the problem is hard to approximate (Corollary 3.5).…”

“…Kaufmann and Wiese [11] considered the graph-drawing problem 1BEND-POINTSETEMBEDDABILITY that will turn out to be related to MINSHIFTEDVER-TICES. They defined a planar graph G = (V , E) to be k-bend embeddable if, for any set S of |V | points in the plane, there is a one-to-one correspondence between V and S that can be extended to a plane drawing of G with at most k bends per edge.…”

Book Embeddability of Series–Parallel Digraphs

“…Bose [5] extends this result by presenting an O(n log 3 n)-time algorithm for computing straightline point-set embeddings of outerplanar graphs on any set of n distinct points in the plane. Since outerplanar graphs are the largest class of graphs admitting a straight-line point-set embedding on any set of points [11], Kaufmann and Wiese [17] investigate the problem of computing a point-set embedding of a planar graph with a small number of bends per edge. They show that if a graph is sub-hamiltonian then it admits a point-set embedding with at most one bend per edge on any set of points in the plane.…”

Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge

“…In order to better locate our contribution in this literature framework, we only recall those results that are more closely related to those of this paper. Key references about k-colored point-set embeddings are the works by Kaufmann and Wiese [19], by Halton [16] and by Pach and Wenger [21]. Kaufmann and Wiese [19] study the monochromatic version of the problem (i.e., the case when k = 1) and prove that a planar graph with n vertices always admits a point-set embedding with at most two bends per edge on any set of n distinct points in the plane; in the same paper, it is also proved that two bends per edge are necessary for some planar graphs and some configurations of points.…”

“…Key references about k-colored point-set embeddings are the works by Kaufmann and Wiese [19], by Halton [16] and by Pach and Wenger [21]. Kaufmann and Wiese [19] study the monochromatic version of the problem (i.e., the case when k = 1) and prove that a planar graph with n vertices always admits a point-set embedding with at most two bends per edge on any set of n distinct points in the plane; in the same paper, it is also proved that two bends per edge are necessary for some planar graphs and some configurations of points. Halton [16] studies the n-chromatic version of the problem (i.e., the case when k = n) and shows that every n-colored planar graphs admits an n-colored point-set embedding on any n-colored set of points; however it does not address the question of minimizing the number of bends per edge in the computed drawing.…”