663 research outputs found
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
Diameter and Treewidth in Minor-Closed Graph Families
It is known that any planar graph with diameter D has treewidth O(D), and
this fact has been used as the basis for several planar graph algorithms. We
investigate the extent to which similar relations hold in other graph families.
We show that treewidth is bounded by a function of the diameter in a
minor-closed family, if and only if some apex graph does not belong to the
family. In particular, the O(D) bound above can be extended to bounded-genus
graphs. As a consequence, we extend several approximation algorithms and exact
subgraph isomorphism algorithms from planar graphs to other graph families.Comment: 15 pages, 12 figure
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Optimal Coalition Structures in Cooperative Graph Games
Representation languages for coalitional games are a key research area in
algorithmic game theory. There is an inherent tradeoff between how general a
language is, allowing it to capture more elaborate games, and how hard it is
computationally to optimize and solve such games. One prominent such language
is the simple yet expressive Weighted Graph Games (WGGs) representation [14],
which maintains knowledge about synergies between agents in the form of an edge
weighted graph.
We consider the problem of finding the optimal coalition structure in WGGs.
The agents in such games are vertices in a graph, and the value of a coalition
is the sum of the weights of the edges present between coalition members. The
optimal coalition structure is a partition of the agents to coalitions, that
maximizes the sum of utilities obtained by the coalitions. We show that finding
the optimal coalition structure is not only hard for general graphs, but is
also intractable for restricted families such as planar graphs which are
amenable for many other combinatorial problems. We then provide algorithms with
constant factor approximations for planar, minor-free and bounded degree
graphs.Comment: 16 pages. A short version of this paper is to appear at AAAI 201
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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