9,130 research outputs found
Recovering sparse graphs
We construct a fixed parameter algorithm parameterized by d and k that takes
as an input a graph G' obtained from a d-degenerate graph G by complementing on
at most k arbitrary subsets of the vertex set of G and outputs a graph H such
that G and H agree on all but f(d,k) vertices.
Our work is motivated by the first order model checking in graph classes that
are first order interpretable in classes of sparse graphs. We derive as a
corollary that if G_0 is a graph class with bounded expansion, then the first
order model checking is fixed parameter tractable in the class of all graphs
that can obtained from a graph G from G_0 by complementing on at most k
arbitrary subsets of the vertex set of G; this implies an earlier result that
the first order model checking is fixed parameter tractable in graph classes
interpretable in classes of graphs with bounded maximum degree
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
We introduce and develop a theory of limits for sequences of sparse graphs
based on graphons, which generalizes both the existing theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.Comment: 44 page
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Respondent driven sampling and sparse graph convergence
We consider a particular respondent-driven sampling procedure governed by a
graphon. By a specific clumping procedure of the sampled vertices we construct
a sequence of sparse graphs. If the sequence of the vertex-sets is stationary
then the sequence of sparse graphs converge to the governing graphon in the
cut-metric. The tools used are concentration inequality for Markov chains and
the Stein-Chen method.Comment: 13 page
Linear Choosability of Sparse Graphs
We study the linear list chromatic number, denoted \lcl(G), of sparse
graphs. The maximum average degree of a graph , denoted \mad(G), is the
maximum of the average degrees of all subgraphs of . It is clear that any
graph with maximum degree satisfies \lcl(G)\ge
\ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if
\mad(G)<12/5 and , then \lcl(G)=\ceil{\Delta(G)/2}+1, and
we give an infinite family of examples to show that this result is best
possible; (2) if \mad(G)<3 and , then
\lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to
show that the bound on \mad(G) cannot be increased in general; (3) if is
planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure
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