79,788 research outputs found
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
Parameterized lower bound and NP-completeness of some -free Edge Deletion problems
For a graph , the -free Edge Deletion problem asks whether there exist
at most edges whose deletion from the input graph results in a graph
without any induced copy of . We prove that -free Edge Deletion is
NP-complete if is a graph with at least two edges and has a component
with maximum number of vertices which is a tree or a regular graph.
Furthermore, we obtain that these NP-complete problems cannot be solved in
parameterized subexponential time, i.e., in time ,
unless Exponential Time Hypothesis fails.Comment: 15 pages, COCOA 15 accepted pape
On the (non-)existence of polynomial kernels for Pl-free edge modification problems
Given a graph G = (V,E) and an integer k, an edge modification problem for a
graph property P consists in deciding whether there exists a set of edges F of
size at most k such that the graph H = (V,E \vartriangle F) satisfies the
property P. In the P edge-completion problem, the set F of edges is constrained
to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no
constraint is imposed on F in the P edge-edition problem. A number of
optimization problems can be expressed in terms of graph modification problems
which have been extensively studied in the context of parameterized complexity.
When parameterized by the size k of the edge set F, it has been proved that if
P is an hereditary property characterized by a finite set of forbidden induced
subgraphs, then the three P edge-modification problems are FPT. It was then
natural to ask whether these problems also admit a polynomial size kernel.
Using recent lower bound techniques, Kratsch and Wahlstrom answered this
question negatively. However, the problem remains open on many natural graph
classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom
asked whether the result holds when the forbidden subgraphs are paths or cycles
and pointed out that the problem is already open in the case of P4-free graphs
(i.e. cographs). This paper provides positive and negative results in that line
of research. We prove that parameterized cograph edge modification problems
have cubic vertex kernels whereas polynomial kernels are unlikely to exist for
the Pl-free and Cl-free edge-deletion problems for large enough l
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
Simultaneous Feedback Vertex Set: A Parameterized Perspective
Given a family of graphs , a graph , and a positive integer
, the -Deletion problem asks whether we can delete at most
vertices from to obtain a graph in . -Deletion
generalizes many classical graph problems such as Vertex Cover, Feedback Vertex
Set, and Odd Cycle Transversal. A graph ,
where the edge set of is partitioned into color classes, is called
an -edge-colored graph. A natural extension of the
-Deletion problem to edge-colored graphs is the
-Simultaneous -Deletion problem. In the latter problem, we
are given an -edge-colored graph and the goal is to find a set
of at most vertices such that each graph , where and , is in . In this work, we
study -Simultaneous -Deletion for being the
family of forests. In other words, we focus on the -Simultaneous
Feedback Vertex Set (-SimFVS) problem. Algorithmically, we show that,
like its classical counterpart, -SimFVS parameterized by is
fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed
constant . In particular, we give an algorithm running in time and a kernel with vertices. The
running time of our algorithm implies that -SimFVS is FPT even when
. We complement this positive result by showing that for
, where is the number of vertices in the input graph,
-SimFVS becomes W[1]-hard. Our positive results answer one of the open
problems posed by Cai and Ye (MFCS 2014)
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