4,278 research outputs found
Incompressibility of H-Free Edge Modification Problems: Towards a Dichotomy
Given a graph G and an integer k, the H-free Edge Editing problem is to find whether there exist at most k pairs of vertices in G such that changing the adjacency of the pairs in G results in a graph without any induced copy of H. The existence of polynomial kernels for H-free Edge Editing (that is, whether it is possible to reduce the size of the instance to k^O(1) in polynomial time) received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs H with at most 4 vertices (e.g., path on 3 or 4 vertices, diamond, paw), but starting from 5 vertices, polynomial kernels are known only if H is either complete or empty. This suggests the conjecture that there is no other H with at least 5 vertices were H-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set ? of nine 5-vertex graphs such that if for every H ? ?, H-free Edge Editing is incompressible and the complexity assumption NP ? coNP/poly holds, then H-free Edge Editing is incompressible for every graph H with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of H-free Edge Editing for every H with at least 5 vertices.
We obtain similar result also for H-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs H where the problem is trivial (graphs with exactly one edge). We obtain a larger set ? of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of H-free Edge Deletion for every graph H with at least 5 vertices. Analogous results follow also for the H-free Edge Completion problem by simple complementation
Parameterized Lower Bound and Improved Kernel for Diamond-free Edge Deletion
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks to find whether there exist at most k edges in the input graph whose deletion results in a diamond-free graph. The problem was proved to be NP-complete and a polynomial kernel of O(k^4) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of O(k^3) vertices for Diamond-free Edge Deletion. We give an alternative proof of the NP-completeness of the problem and observe that it cannot be solved in time 2^{o(k)} * n^{O(1)}, unless the Exponential Time Hypothesis fails
Contracting edges to destroy a pattern: A complexity study
Given a graph G and an integer k, the objective of the -Contraction
problem is to check whether there exists at most k edges in G such that
contracting them in G results in a graph satisfying the property . We
investigate the problem where is `H-free' (without any induced copies of
H). It is trivial that H-free Contraction is polynomial-time solvable if H is a
complete graph of at most two vertices. We prove that, in all other cases, the
problem is NP-complete. We then investigate the fixed-parameter tractability of
these problems. We prove that whenever H is a tree, except for seven trees,
H-free Contraction is W[2]-hard. This result along with the known results
leaves behind three unknown cases among trees.Comment: 30 pages, 10 figures, a short version is accepted to FCT 202
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
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
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