11 research outputs found
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time
Thomassen characterized some 1-plane embedding as the forbidden configuration
such that a given 1-plane embedding of a graph is drawable in straight-lines if
and only if it does not contain the configuration [C. Thomassen, Rectilinear
drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988].
In this paper, we characterize some 1-plane embedding as the forbidden
configuration such that a given 1-plane embedding of a graph can be re-embedded
into a straight-line drawable 1-plane embedding of the same graph if and only
if it does not contain the configuration. Re-embedding of a 1-plane embedding
preserves the same set of pairs of crossing edges.
We give a linear-time algorithm for finding a straight-line drawable 1-plane
re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016). This is an extended
abstract. For a full version of this paper, see Hong S-H, Nagamochi H.:
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time,
Technical Report TR 2016-002, Department of Applied Mathematics and Physics,
Kyoto University (2016
3D Visibility Representations of 1-planar Graphs
We prove that every 1-planar graph G has a z-parallel visibility
representation, i.e., a 3D visibility representation in which the vertices are
isothetic disjoint rectangles parallel to the xy-plane, and the edges are
unobstructed z-parallel visibilities between pairs of rectangles. In addition,
the constructed representation is such that there is a plane that intersects
all the rectangles, and this intersection defines a bar 1-visibility
representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs
We propose a fixed-parameter tractable algorithm for the \textsc{Max-Cut}
problem on embedded 1-planar graphs parameterized by the crossing number of
the given embedding. A graph is called 1-planar if it can be drawn in the plane
with at most one crossing per edge. Our algorithm recursively reduces a
1-planar graph to at most planar graphs, using edge removal and node
contraction. The \textsc{Max-Cut} problem is then solved on the planar graphs
using established polynomial-time algorithms. We show that a maximum cut in the
given 1-planar graph can be derived from the solutions for the planar graphs.
Our algorithm computes a maximum cut in an embedded 1-planar graph with
nodes and edge crossings in time .Comment: conference version from IWOCA 201
A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs
We prove a theorem that for an integer s >= 0, if 12s + 7 is a prime number, then the. number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K-n, where it = (12s + 7) (6s + 7), is at least 2(n3/2(root 2/72+o(1))). By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least 2(alpha nl+o(nl)), l > 1, nonisomorphic nonorientable triangular embeddings of K-n for it =6t + 1, t equivalent to 2 mod 3. To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given. (C) 2007 Elsevier B.V. All rights reserved.X11sciescopu
NONORIENTABLE TRIANGULAR EMBEDDINGS OF COMPLETE GRAPHS WITH ARBITRARILY LARGE LOOSENESS
The looseness of a triangular embedding of a complete graph in a closed surface is the minimum integer m such that for every assignment of m colors to the vertices of the embedding (such that all m colors are used) there is a face incident with vertices of three distinct colors. In this paper we show that for every p >= 3 there is a nonorientable triangular embedding of a complete graph with looseness at least p. (C) 2007 Elsevier B.V. All rights reserved.X11sciescopu
Ortho-Polygon Visibility Representations of Embedded Graphs
An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. Namely, we prove that if G is 1-plane (i.e., it has at most one crossing per edge) an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute in O(n) time an OPVR of G whose vertex complexity is bounded by a constant. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed. Finally, we present the results of an experimental study on the vertex complexity of OPVRs of 1-plane graphs