59 research outputs found
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs
A graph is called a strong (resp. weak) bar 1-visibility graph if its
vertices can be represented as horizontal segments (bars) in the plane so that
its edges are all (resp. a subset of) the pairs of vertices whose bars have a
-thick vertical line connecting them that intersects at most one
other bar.
We explore the relation among weak (resp. strong) bar 1-visibility graphs and
other nearly planar graph classes. In particular, we study their relation to
1-planar graphs, which have a drawing with at most one crossing per edge;
quasi-planar graphs, which have a drawing with no three mutually crossing
edges; the squares of planar 1-flow networks, which are upward digraphs with
in- or out-degree at most one. Our main results are that 1-planar graphs and
the (undirected) squares of planar 1-flow networks are weak bar 1-visibility
graphs and that these are quasi-planar graphs
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
Parameterized Complexity of 1-Planarity
We consider the problem of finding a 1-planar drawing for a general graph,
where a 1-planar drawing is a drawing in which each edge participates in at
most one crossing. Since this problem is known to be NP-hard we investigate the
parameterized complexity of the problem with respect to the vertex cover
number, tree-depth, and cyclomatic number. For these parameters we construct
fixed-parameter tractable algorithms. However, the problem remains NP-complete
for graphs of bounded bandwidth, pathwidth, or treewidth.Comment: WADS 201
Upward Book Embeddings of st-Graphs
We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE
Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size
In the ?-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G-S contains no graph from the finite family ? as a minor. It is known that whenever ? contains at least one planar graph, then ?-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size k^{?(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size.
We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of ?-Minor-Free Deletion for the family ? = {K?, K_{2,3}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with ?(k?) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has ?(k?) vertices and edges
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