Theoretical results on at most 1-bend embeddability of graphs (original) (raw)

Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Journal of Graph Algorithms and Applications, 2002

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.

Clin D'oeil on L1-embeddable Planar Graphs

Discrete Applied Mathematics, 1997

In this note we present some properties of LI-embeddable planar graphs. We present a characterization of graphs isometrically embeddable into half-cubes. This result implies that every planar Li-graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of a planar Li-graph G the subgraph H of G bounded by C is also Li-embeddable.

Miscellaneous properties of embeddings of line, total and middle graphs

Discrete Mathematics, 2001

proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that R(G) is planar, where R is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not R(G) is planar in an O(n) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The ÿrst characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs.

Windrose planarity: embedding graphs with direction-constrained edges

Symposium on Discrete Algorithms, 2016

Given a planar graph G(V, E) and a partition of the neighbors of each vertex v ∈ V in four sets v, v, v, and v, the problem WINDROSE PLANARITY asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v is above and to the right of v, (ii) each neighbor u ∈ v is above and to the left of v, (iii) each neighbor u ∈ v is below and to the left of v, (iv) each neighbor u ∈ v is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is N P-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a combinatorial embedding that is given as part of the input. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph admitting a windrose-planar drawing we show how to construct one with at most one bend per edge on an O(n) × O(n) grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

Existence of Polyhedral Embeddings of Graphs

Combinatorica, 2001

It is proved that the decision problem about the existence of an embedding of face-width 3 of a given graph is NP-complete. A similar result is proved for some related decision problems. This solves a problem raised by Neil Robertson.

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

Discrete & Computational Geometry, 2010

This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.

The point-set embeddability problem for plane graphs

Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12, 2012

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? 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 embedding that have constant treewidth and constant face-degree. We also 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.

Universal Sets of n Points for 1-Bend Drawings of Planar Graphs with n Vertices

Lecture Notes in Computer Science, 2008

Theoretical results on at most 1-bend embeddability of graphs (original) (raw)

Related papers