96 research outputs found
Constructions of Large Graphs on Surfaces
We consider the degree/diameter problem for graphs embedded in a surface,
namely, given a surface and integers and , determine the
maximum order of a graph embeddable in with
maximum degree and diameter . We introduce a number of
constructions which produce many new largest known planar and toroidal graphs.
We record all these graphs in the available tables of largest known graphs.
Given a surface of Euler genus and an odd diameter , the
current best asymptotic lower bound for is given by
Our constructions produce
new graphs of order \begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if
$\Sigma$ is the Klein bottle}\\
\(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}&
\text{otherwise,}\end{cases} thus improving the former value by a factor of
4.Comment: 15 pages, 7 figure
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
Minor Excluded Network Families Admit Fast Distributed Algorithms
Distributed network optimization algorithms, such as minimum spanning tree,
minimum cut, and shortest path, are an active research area in distributed
computing. This paper presents a fast distributed algorithm for such problems
in the CONGEST model, on networks that exclude a fixed minor.
On general graphs, many optimization problems, including the ones mentioned
above, require rounds of communication in the CONGEST
model, even if the network graph has a much smaller diameter. Naturally, the
next step in algorithm design is to design efficient algorithms which bypass
this lower bound on a restricted class of graphs. Currently, the only known
method of doing so uses the low-congestion shortcut framework of Ghaffari and
Haeupler [SODA'16]. Building off of their work, this paper proves that excluded
minor graphs admit high-quality shortcuts, leading to an round
algorithm for the aforementioned problems, where is the diameter of the
network graph. To work with excluded minor graph families, we utilize the Graph
Structure Theorem of Robertson and Seymour. To the best of our knowledge, this
is the first time the Graph Structure Theorem has been used for an algorithmic
result in the distributed setting.
Even though the proof is involved, merely showing the existence of good
shortcuts is sufficient to obtain simple, efficient distributed algorithms. In
particular, the shortcut framework can efficiently construct near-optimal
shortcuts and then use them to solve the optimization problems. This, combined
with the very general family of excluded minor graphs, which includes most
other important graph classes, makes this result of significant interest
A Note on the Maximum Genus of Graphs with Diameter 4
Let G be a simple graph with diameter four, if G does not contain complete
subgraph K3 of order three
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
Extremal density for sparse minors and subdivisions
We prove an asymptotically tight bound on the extremal density guaranteeing
subdivisions of bounded-degree bipartite graphs with a mild separability
condition. As corollaries, we answer several questions of Reed and Wood on
embedding sparse minors. Among others,
average degree is sufficient to force the
grid as a topological minor;
average degree forces every -vertex planar graph
as a minor, and the constant is optimal, furthermore, surprisingly, the
value is the same for -vertex graphs embeddable on any fixed surface;
a universal bound of on average degree forcing every
-vertex graph in any nontrivial minor-closed family as a minor, and the
constant 2 is best possible by considering graphs with given treewidth.Comment: 33 pages, 6 figure
Every Minor-Closed Property of Sparse Graphs is Testable
Suppose is a graph with degrees bounded by , and one needs to remove
more than of its edges in order to make it planar. We show that in
this case the statistics of local neighborhoods around vertices of is far
from the statistics of local neighborhoods around vertices of any planar graph
with the same degree bound. In fact, a similar result is proved for any
minor-closed property of bounded degree graphs.
As an immediate corollary of the above result we infer that many well studied
graph properties, like being planar, outer-planar, series-parallel, bounded
genus, bounded tree-width and several others, are testable with a constant
number of queries, where the constant may depend on and , but not
on the graph size. None of these properties was previously known to be testable
even with queries
Minors in expanding graphs
Extending several previous results we obtained nearly tight estimates on the
maximum size of a clique-minor in various classes of expanding graphs. These
results can be used to show that graphs without short cycles and other H-free
graphs contain large clique-minors, resolving some open questions in this area
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