Point-set Embeddability Research Articles

Grid obstacle representation of graphs

The grid obstacle representation, or alternately, ℓ1-obstacle representation of a graph G=(V,E) is an injective function f:V→Z2 and a set of point obstacles O on the grid points of Z2 (where no vertex of V has been mapped) such that uv is an edge in G if and only if there exists a Manhattan path between f(u) and f(v) in Z2 avoiding the obstacles of O and points in f(V). This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation. Moreover, we show that every graph admits a grid obstacle representation in Z3. We also show NP-hardness result for the point set embeddability of an ℓ1-obstacle representation.

On the curve complexity of 3-colored point-set embeddings

We establish new results on the curve complexity of k-colored point-set embeddings when k=3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require Ω(n13) bends on Ω(n23) edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k-colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has O(1) independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k-colored simultaneous embeddability problem and study its relationship with the k-colored point-set embeddability problem.

Optimal point-set embedding of wheel graphs and a sub-class of 3-trees

In the point-set embeddability problem, given a planar graph and a point set in the plane, one asks if there exists a straight-line drawing of the graph in the plane such that the nodes of the graph are mapped one-to-one onto the points of the set. By optimal embedding of a graph, we mean an embedding of minimum length or area. The point-set embeddability problem is NP-complete, and so far it has been solved polynomially only for a few classes of planar graphs. In this paper, we present optimal \(O(n\log {n})\)-time algorithms for embedding and \(O(n^2)\)-time algorithms for optimal embedding of wheel graphs and a sub-class of 3-trees.

Point-Set Embedding in Three Dimensions

Given a graph $G$ with $n$ vertices and $m$ edges, and a set $P$ of $n$ points on a three-dimensional integer grid, the 3D Point-Set Embeddability problem is to determine a (three-dimensional) crossing-free drawing of $G$ with vertices located at $P$ and with edges drawn as poly-lines with bend-points at integer grid points. We solve a variant of the problem in which the points of $P$ lie on a plane. The resulting drawing lies in a bounding box of reasonable volume and uses at most $O(\log m)$ bends per edge. If a particular point-set $P$ is not specified, we show that the graph $G$ can be drawn crossing-free with at most $O(\log m)$ bends per edge in a volume bounded by $O((n+m) \log m)$. Our construction is asymptotically similar to previously known drawings, however avoids a possibly non-polynomial preprocessing step.

Planar embedding of trees on point sets without the general position assumption

The problem of point-set embedding of a planar graph $G$ on a point set $P$ in the plane is defined as finding a straight-line planar drawing of $G$ such that the nodes of $G$ are mapped one to one on the points of $P$. Previous works in this area mostly assume that the points of $P$ are in general position, i.e. $P$ does not contain any three collinear points. However, in most of the real applications we cannot assume the general position assumption. In this paper, we show that deciding the point-set embeddability of trees without the general position assumption is NP-complete. Then we introduce an algorithm for point-set embedding of $n$-node binary trees with at most $\frac{n}{3}$ total bends on any point set. We also give some results when the problem is limited to degree-constrained trees and point sets having constant number of collinear points.

Computing upward topological book embeddings of upward planar digraphs

We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.

THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

On upward point set embeddability

We study the problem of upward point set embeddability, that is the problem to decide whether an n-vertex directed graph has an upward planar drawing when its vertices have to be placed on the points of a given point set of size n. We first present some positive and negative results concerning directed trees and convex point sets. Next, we prove that upward point set embeddability can be solved in polynomial time for the case of a directed tree and a convex point set. Further, we extend our approach to the class of outerplanar directed graphs. This implies that upward point set embeddability can be efficiently solved for the case of convex point sets. Finally, we show that the general problem of upward point set embeddability is NP-complete even for 2-convex point sets.

IMPROVED ALGORITHMS FOR THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE 3-TREES

In the point-set embeddability problem, we are given a plane graph G with n vertices and a point set S with the same number of points. Now the goal is to answer the question whether there exists a straight-line drawing of G such that each vertex is represented as a distinct point of S as well as to provide an embedding if one does exist. This problem has recently been solved in O(n2 log n) time for plane 3-trees. In this paper, we present a new efficient algorithm with time complexity O(n4/3+ϵ log n). We also present an O(nk4) time algorithm for the case when |S| = k > n. This is a significant improvement over the best algorithm for this case in the literature, which runs in O(nk8) time.

Point-set embeddings of plane 3-trees

A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight line segment. Given a set S of n points in the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits a point-set embedding on S is NP-complete in general and even when G is 2-connected and 2-outerplanar. In this paper, we give an O ( n 2 ) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω ( n log n ) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. We then consider a variant of the problem, where we are given a plane 3-tree G with n vertices and a set S of k > n points, and present a dynamic programming algorithm to find a point-set embedding of G on S if it exists. Furthermore, we show that the point-set embeddability problem for planar partial 3-trees is also NP-complete.

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

Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts. We exploit the interplay between these notions to describe colored sets of points and to design polynomial-time algorithms to embed k-colored planar graphs on these sets such that the resulting drawings have $\mathcal{O}(k)$bends per edge.

Book Embeddability of Series–Parallel Digraphs

In this paper we deal with the problem of computing upward two-page book embeddings of Two Terminal Series-Parallel (TTSP) digraphs, which are a subclass of series-parallel digraphs. An optimal O(n) time and space algorithm to compute an upward two-page book embedding of a TTSP-digraph with n vertices is presented. A previous algorithm of Alzohairi and Rival [1] runs in O(n3) time and assumes that the input series-parallel digraph does not have transitive edges. An application of this result to a computational geometry problem is also discussed. More precisely, upward two-page book embeddings are used to deal with the upward point-set embeddability problem, i.e., the problem of mapping planar digraphs onto a given set of points in the plane so that all edges are monotonically increasing in a common direction. The equivalence between upward two-page book embeddability and upward point-set embeddability with at most one bend per edge on any given set of points is proved. An O(n log n)-time algorithm for computing an upward point-set embedding with at most one bend per edge for TTSP-digraphs is presented.

Curve-constrained drawings of planar graphs

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ∈ C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented.