19 research outputs found
On Polynomial Kernelization of H-free Edge Deletion
For a set H of graphs, the H-free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some k∈N, whose deletion results in a graph without an induced copy of any of the graphs in H . The problem is known to be fixed-parameter tractable if H is of finite cardinality. In this paper, we present a polynomial kernel for this problem for any fixed finite set H of connected graphs for the case where the input graphs are of bounded degree. We use a single kernelization rule which deletes vertices ‘far away’ from the induced copies of every H∈H in the input graph. With a slightly modified kernelization rule, we obtain polynomial kernels for H-free Edge Deletion under the following three settings
Editing to a Graph of Given Degrees
We consider the Editing to a Graph of Given Degrees problem that asks for a
graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d},
whether it is possible to obtain a graph G' from G such that the degree of v is
\delta(v) for any vertex v by at most k vertex or edge deletions or edge
additions. We construct an FPT-algorithm for Editing to a Graph of Given
Degrees parameterized by d+k. We complement this result by showing that the
problem has no polynomial kernel unless NP\subseteq coNP/poly
On Polynomial Kernelization of -free Edge Deletion
For a set of graphs , the \textsc{-free Edge
Deletion} problem asks to find whether there exist at most edges in the
input graph whose deletion results in a graph without any induced copy of
. In \cite{cai1996fixed}, it is shown that the problem is
fixed-parameter tractable if is of finite cardinality. However,
it is proved in \cite{cai2013incompressibility} that if is a
singleton set containing , for a large class of , there exists no
polynomial kernel unless . In this paper, we present a
polynomial kernel for this problem for any fixed finite set of
connected graphs and when the input graphs are of bounded degree. We note that
there are \textsc{-free Edge Deletion} problems which remain
NP-complete even for the bounded degree input graphs, for example
\textsc{Triangle-free Edge Deletion}\cite{brugmann2009generating} and
\textsc{Custer Edge Deletion(-free Edge
Deletion)}\cite{komusiewicz2011alternative}. When contains
, we obtain a stronger result - a polynomial kernel for -free
input graphs (for any fixed ). We note that for , there is an
incompressibility result for \textsc{-free Edge Deletion} for general
graphs \cite{cai2012polynomial}. Our result provides first polynomial kernels
for \textsc{Claw-free Edge Deletion} and \textsc{Line Edge Deletion} for
-free input graphs which are NP-complete even for -free
graphs\cite{yannakakis1981edge} and were raised as open problems in
\cite{cai2013incompressibility,open2013worker}.Comment: 12 pages. IPEC 2014 accepted pape
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