2,900 research outputs found
Stack and Queue Layouts via Layered Separators
It is known that every proper minor-closed class of graphs has bounded
stack-number (a.k.a. book thickness and page number). While this includes
notable graph families such as planar graphs and graphs of bounded genus, many
other graph families are not closed under taking minors. For fixed and ,
we show that every -vertex graph that can be embedded on a surface of genus
with at most crossings per edge has stack-number ;
this includes -planar graphs. The previously best known bound for the
stack-number of these families was , except in the case
of -planar graphs. Analogous results are proved for map graphs that can be
embedded on a surface of fixed genus. None of these families is closed under
taking minors. The main ingredient in the proof of these results is a
construction proving that -vertex graphs that admit constant layered
separators have stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
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
On Counting Oracles for Path Problems
We initiate the study of counting oracles for various path problems in graphs. Distance oracles have gained a lot of attention in recent years, with studies of the underlying space and time tradeoffs. For a given graph G, a distance oracle is a data structure which can be used to answer distance queries for pairs of vertices s,t in V(G). In this work, we extend the set up to answering counting queries: for a pair of vertices s,t, the oracle needs to provide the number of (shortest or all) paths from s to t. We present O(n^{1.5}) preprocessing time, O(n^{1.5}) space, and O(sqrt{n}) query time algorithms for oracles counting shortest paths in planar graphs and for counting all paths in planar directed acyclic graphs. We extend our results to other graphs which admit small balanced separators and present applications where our oracle improves the currently best known running times
Engineering planar separator algorithms
We consider classical linear-time planar separator
algorithms, determining for a given planar graph a
small subset of the nodes whose removal separates the
graph into two components of similar size. These algorithms
are based upon Planar Separator Theorems, which
guarantee separators of size asymptotically in the
square root of the number of nodes n and remaining
components of size less than 2n/3. In this work, we
present a comprehensive experimental study of the
algorithms applied to a large variety of graphs, where
the main goal is to find separators that do not only
satisfy upper bounds but also possess other desirable
qualities with respect to separator size and component
balance. We propose the usage of fundamental cycles,
whose size is at most twice the diameter of the graph, as planar
separators: For graphs of small diameter the
guaranteed bound is better than the bounds of the classical
algorithms, and it turns out that this simple strategy almost
always outperforms the other algorithms, even for graphs with
large diameter
Engineering Planar Separator Algorithms
We consider classical linear-time planar separator
algorithms, determining for a given planar graph a
small subset of the nodes whose removal separates the
graph into two components of similar size. These algorithms
are based upon Planar Separator Theorems, which
guarantee separators of size asymptotically in the
square root of the number of nodes n and remaining
components of size less than 2n/3. In this work, we
present a comprehensive experimental study of the
algorithms applied to a large variety of graphs, where
the main goal is to find separators that do not only
satisfy upper bounds but also possess other desirable
qualities with respect to separator size and component
balance. We propose the usage of fundamental cycles,
whose size is at most twice the diameter of the graph, as planar
separators: For graphs of small diameter the
guaranteed bound is better than the bounds of the classical
algorithms, and it turns out that this simple strategy almost
always outperforms the other algorithms, even for graphs with
large diameter
Crossing Patterns in Nonplanar Road Networks
We define the crossing graph of a given embedded graph (such as a road
network) to be a graph with a vertex for each edge of the embedding, with two
crossing graph vertices adjacent when the corresponding two edges of the
embedding cross each other. In this paper, we study the sparsity properties of
crossing graphs of real-world road networks. We show that, in large road
networks (the Urban Road Network Dataset), the crossing graphs have connected
components that are primarily trees, and that the remaining non-tree components
are typically sparse (technically, that they have bounded degeneracy). We prove
theoretically that when an embedded graph has a sparse crossing graph, it has
other desirable properties that lead to fast algorithms for shortest paths and
other algorithms important in geographic information systems. Notably, these
graphs have polynomial expansion, meaning that they and all their subgraphs
have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems(ACM
SIGSPATIAL 2017
Edge separators for graphs of bounded genus with applications
-vertex graph of positive genus and maximal degree has an edge separator. This bound is best possible to within a constant factor. The separator can be found in time provided that we start with an imbedding of the graph in its genus surface. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric problem, to efficient embeddings of graphs of genus into various classes of graphs including trees, meshes and hypercubes and to showing lower bounds on crossing numbers of and drawn on surfaces of genus
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