18,709 research outputs found
Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs
In this paper we deal with the problem of computing the exact crossing number
of almost planar graphs and the closely related problem of computing the exact
anchored crossing number of a pair of planar graphs. It was shown by [Cabello
and Mohar, 2013] that both problems are NP-hard; although they required an
unbounded number of high-degree vertices (in the first problem) or an unbounded
number of anchors (in the second problem) to prove their result. Somehow
surprisingly, only three vertices of degree greater than 3, or only three
anchors, are sufficient to maintain hardness of these problems, as we prove
here. The new result also improves the previous result on hardness of joint
crossing number on surfaces by [Hlin\v{e}n\'y and Salazar, 2015]. Our result is
best possible in the anchored case since the anchored crossing number of a pair
of planar graphs with two anchors each is trivial, and close to being best
possible in the almost planar case since the crossing number is efficiently
computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello
and Mohar 2011]
Konstruksi Famili Graf Hampir Planar dengan Angka Perpotongan Tertentu
KONSTRUKSI FAMILI GRAF HAMPIR PLANAR DENGAN ANGKA PERPOTONGAN TERTENTU Benny Pinontoan1) 1) Program Studi Matematika FMIPA Universitas Sam Ratulangi Manado, 95115ABSTRAK Sebuah graf adalah pasangan himpunan tak kosong simpul dan himpunan sisi. Graf dapat digambar pada bidang dengan atau tanpa perpotongan. Angka perpotongan adalah jumlah perpotongan terkecil di antara semua gambar graf pada bidang. Graf dengan angka perpotongan nol disebut planar. Graf memiliki penerapan penting pada desain Very Large Scale of Integration (VLSI). Sebuah graf dinamakan perpotongan kritis jika penghapusan sebuah sisi manapun menurunkan angka perpotongannya, sedangkan sebuah graf dinamakan hampir planar jika menghapus salah satu sisinya membuat graf yang sisa menjadi planar. Banyak famili graf perpotongan kritis yang dapat dibentuk dari bagian-bagian kecil yang disebut ubin yang diperkenalkan oleh Pinontoan dan Richter (2003). Pada tahun 2010, Bokal memperkenalkan operasi perkalian zip untuk graf. Dalam artikel ini ditunjukkan sebuah konstruksi dengan menggunakan ubin dan perkalian zip yang jika diberikan bilangan bulat k ³ 1, dapat menghasilkan famili tak hingga graf hampir planar dengan angka perpotongan k. Kata kunci: angka perpotongan, ubin graf, graf hampir planar. CONSTRUCTION OF INFINITE FAMILIES OF ALMOST PLANAR GRAPH WITH GIVEN CROSSING NUMBER ABSTRACT A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossings among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. Crossing number problems find important applications in the design of layout of Very Large Scale of Integration (VLSI). A graph is crossing-critical if deleting of any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. Many infinite sequences of crossing-critical graphs can be made up by gluing small pieces, called tiles introduced by Pinontoan and Richter (2003). In 2010, Bokal introduced the operation zip product of graphs. This paper shows a construction by using tiles and zip product, given an integer k ³ 1, to build an infinite family of almost planar graphs having crossing number k
CONSTRUCTION OF INFINITE FAMILIES OF ALMOST PLANAR GRAPH WITH GIVEN CROSSING NUMBER
A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossings among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. Crossing number problems find important applications in the design of layout of Very Large Scale of Integration (VLSI). A graph is crossing-critical if deleting of any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. Many infinite sequences of crossing-critical graphs can be made up by gluing small pieces, called tiles introduced b
Untangling Circular Drawings: Algorithms and Complexity
We consider the problem of untangling a given (non-planar) straight-line
circular drawing of an outerplanar graph into a planar
straight-line circular drawing by shifting a minimum number of vertices to a
new position on the circle. For an outerplanar graph , it is clear that such
a crossing-free circular drawing always exists and we define the circular
shifting number shift as the minimum number of vertices that are
required to be shifted in order to resolve all crossings of . We show
that the problem Circular Untangling, asking whether shift
for a given integer , is NP-complete. For -vertex outerplanar graphs, we
obtain a tight upper bound of shift. Based on these results we study Circular Untangling for almost-planar
circular drawings, in which a single edge is involved in all the crossings. In
this case, we provide a tight upper bound shift and present a constructive polynomial-time algorithm to
compute the circular shifting number of almost-planar drawings.Comment: 20 pages, 10 figures, extended version of ISAAC 2021 pape
Universal Geometric Graphs
We introduce and study the problem of constructing geometric graphs that have
few vertices and edges and that are universal for planar graphs or for some
sub-class of planar graphs; a geometric graph is \emph{universal} for a class
of planar graphs if it contains an embedding, i.e., a
crossing-free drawing, of every graph in .
Our main result is that there exists a geometric graph with vertices and
edges that is universal for -vertex forests; this extends to
the geometric setting a well-known graph-theoretic result by Chung and Graham,
which states that there exists an -vertex graph with edges
that contains every -vertex forest as a subgraph. Our bound on
the number of edges cannot be improved, even if more than vertices are
allowed.
We also prove that, for every positive integer , every -vertex convex
geometric graph that is universal for -vertex outerplanar graphs has a
near-quadratic number of edges, namely ; this almost
matches the trivial upper bound given by the -vertex complete
convex geometric graph.
Finally, we prove that there exists an -vertex convex geometric graph with
vertices and edges that is universal for -vertex
caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will
appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2020
Straight-line Drawability of a Planar Graph Plus an Edge
We investigate straight-line drawings of topological graphs that consist of a
planar graph plus one edge, also called almost-planar graphs. We present a
characterization of such graphs that admit a straight-line drawing. The
characterization enables a linear-time testing algorithm to determine whether
an almost-planar graph admits a straight-line drawing, and a linear-time
drawing algorithm that constructs such a drawing, if it exists. We also show
that some almost-planar graphs require exponential area for a straight-line
drawing
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