144 research outputs found
Independent Set Reconfiguration in Cographs
We study the following independent set reconfiguration problem, called
TAR-Reachability: given two independent sets and of a graph , both
of size at least , is it possible to transform into by adding and
removing vertices one-by-one, while maintaining an independent set of size at
least throughout? This problem is known to be PSPACE-hard in general. For
the case that is a cograph (i.e. -free graph) on vertices, we show
that it can be solved in time , and that the length of a shortest
reconfiguration sequence from to is bounded by , if such a
sequence exists.
More generally, we show that if is a graph class for which (i)
TAR-Reachability can be solved efficiently, (ii) maximum independent sets can
be computed efficiently, and which satisfies a certain additional property,
then the problem can be solved efficiently for any graph that can be obtained
from a collection of graphs in using disjoint union and complete join
operations. Chordal graphs are given as an example of such a class
On The Complexity of Distance- Independent Set Reconfiguration
For a fixed positive integer , a distance- independent set
(DIS) of a graph is a vertex subset whose distance between any two members
is at least . Imagine that there is a token placed on each member of a
DIS. Two DISs are adjacent under Token Sliding () if one can
be obtained from the other by moving a token from one vertex to one of its
unoccupied adjacent vertices. Under Token Jumping (), the target
vertex needs not to be adjacent to the original one. The Distance-
Independent Set Reconfiguration (DISR) problem under
asks if there is a corresponding sequence of adjacent
DISs that transforms one given DIS into another. The problem for ,
also known as the Independent Set Reconfiguration problem, has been
well-studied in the literature and its computational complexity on several
graph classes has been known. In this paper, we study the computational
complexity of DISR on different graphs under and
for any fixed . On chordal graphs, we show that DISR under
is in when is even and
-complete when is odd. On split graphs, there is an
interesting complexity dichotomy: DISR is -complete for but in for under , while under
it is in for but -complete
for . Additionally, certain well-known hardness results for on
general graphs, perfect graphs, planar graphs of maximum degree three and
bounded bandwidth can be extended for .Comment: 14 pages, 8 figures, minor revision, to appear in WALCOM 202
On The Complexity of Distance- Independent Set Reconfiguration
This PDF is not the same as the Accepted Paper for 'WALCOM 2023'.For a fixed positive integer , a distance- independent set (DIS) of a graph is a vertex subset whose distance between any two members is at least . Imagine that there is a token placed on each member of a DIS. Two DISs are adjacent under Token Sliding () if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (), the target vertex needs not to be adjacent to the original one. The Distance- Independent Set Reconfiguration (DISR) problem under asks if there is a corresponding sequence of adjacent DISs that transforms one given DIS into another. The problem for , also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DISR on different graphs under and for any fixed . On chordal graphs, we show that DISR under is in when is even and -complete when is odd. On split graphs, there is an interesting complexity dichotomy: DISR is -complete for but in for under , while under it is in for but -complete for . Additionally, certain well-known hardness results for on general graphs, perfect graphs, planar graphs of maximum degeree three and bounded bandwidth can be extended for
Fixed-Parameter Tractability of Token Jumping on Planar Graphs
Suppose that we are given two independent sets and of a graph
such that , and imagine that a token is placed on each vertex in
. The token jumping problem is to determine whether there exists a
sequence of independent sets which transforms into so that each
independent set in the sequence results from the previous one by moving exactly
one token to another vertex. This problem is known to be PSPACE-complete even
for planar graphs of maximum degree three, and W[1]-hard for general graphs
when parameterized by the number of tokens. In this paper, we present a
fixed-parameter algorithm for the token jumping problem on planar graphs, where
the parameter is only the number of tokens. Furthermore, the algorithm can be
modified so that it finds a shortest sequence for a yes-instance. The same
scheme of the algorithms can be applied to a wider class of graphs,
-free graphs for any fixed integer , and it yields
fixed-parameter algorithms
Galactic Token Sliding
International audienc
Shortest Dominating Set Reconfiguration under Token Sliding
In this paper, we present novel algorithms that efficiently compute a
shortest reconfiguration sequence between two given dominating sets in trees
and interval graphs under the Token Sliding model. In this problem, a graph is
provided along with its two dominating sets, which can be imagined as tokens
placed on vertices. The objective is to find a shortest sequence of dominating
sets that transforms one set into the other, with each set in the sequence
resulting from sliding a single token in the previous set. While identifying
any sequence has been well studied, our work presents the first polynomial
algorithms for this optimization variant in the context of dominating sets.Comment: To appear at FCT 2023 (Fundamentals of Computation Theory
Reconfiguring k-path vertex covers
A vertex subset of a graph is called a -path vertex cover if every
path on vertices in contains at least one vertex from . The
\textsc{-Path Vertex Cover Reconfiguration (-PVCR)} problem asks if one
can transform one -path vertex cover into another via a sequence of -path
vertex covers where each intermediate member is obtained from its predecessor
by applying a given reconfiguration rule exactly once. We investigate the
computational complexity of \textsc{-PVCR} from the viewpoint of graph
classes under the well-known reconfiguration rules: ,
, and . The problem for , known as the
\textsc{Vertex Cover Reconfiguration (VCR)} problem, has been well-studied in
the literature. We show that certain known hardness results for \textsc{VCR} on
different graph classes including planar graphs, bounded bandwidth graphs,
chordal graphs, and bipartite graphs, can be extended for \textsc{-PVCR}. In
particular, we prove a complexity dichotomy for \textsc{-PVCR} on general
graphs: on those whose maximum degree is (and even planar), the problem is
-complete, while on those whose maximum degree is (i.e.,
paths and cycles), the problem can be solved in polynomial time. Additionally,
we also design polynomial-time algorithms for \textsc{-PVCR} on trees under
each of and . Moreover, on paths, cycles, and
trees, we describe how one can construct a reconfiguration sequence between two
given -path vertex covers in a yes-instance. In particular, on paths, our
constructed reconfiguration sequence is shortest.Comment: 29 pages, 4 figures, to appear in WALCOM 202
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