5,742 research outputs found
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
On the size of planarly connected crossing graphs
We prove that if an -vertex graph can be drawn in the plane such that
each pair of crossing edges is independent and there is a crossing-free edge
that connects their endpoints, then has edges. Graphs that admit
such drawings are related to quasi-planar graphs and to maximal -planar and
fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
On the size of identifying codes in triangle-free graphs
In an undirected graph , a subset such that is a
dominating set of , and each vertex in is dominated by a distinct
subset of vertices from , is called an identifying code of . The concept
of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in
1998. For a given identifiable graph , let \M(G) be the minimum
cardinality of an identifying code in . In this paper, we show that for any
connected identifiable triangle-free graph on vertices having maximum
degree , \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is
asymptotically tight up to constants due to various classes of graphs including
-ary trees, which are known to have their minimum identifying code
of size . We also provide improved bounds for
restricted subfamilies of triangle-free graphs, and conjecture that there
exists some constant such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c
holds for any nontrivial connected identifiable graph
Notes on large angle crossing graphs
A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges
in G intersect at an angle of at least a. The concept of right angle crossing
(RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown
that any RAC graph with n vertices has at most 4n-10 edges and that there are
infinitely many values of n for which there exists a RAC graph with n vertices
and 4n-10 edges. In this paper, we give upper and lower bounds for the number
of edges in aAC graphs for all 0 < a < Pi/2
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
On the Number of Edges of Fan-Crossing Free Graphs
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1
if there are no k+1 edges , such that have a
common endpoint and crosses all . We prove a tight bound of 4n-8 on
the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9
bound for a straight-edge drawing. For k > 2, we prove an upper bound of
3(k-1)(n-2) edges. We also discuss generalizations to monotone graph
properties
Coloring curves that cross a fixed curve
We prove that for every integer , the class of intersection graphs
of curves in the plane each of which crosses a fixed curve in at least one and
at most points is -bounded. This is essentially the strongest
-boundedness result one can get for this kind of graph classes. As a
corollary, we prove that for any fixed integers and , every
-quasi-planar topological graph on vertices with any two edges crossing
at most times has edges.Comment: Small corrections, improved presentatio
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