46 research outputs found
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
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
The Perfect Matching Reconfiguration Problem
We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time
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
Shortest paths between shortest paths and independent sets
We study problems of reconfiguration of shortest paths in graphs. We prove
that the shortest reconfiguration sequence can be exponential in the size of
the graph and that it is NP-hard to compute the shortest reconfiguration
sequence even when we know that the sequence has polynomial length. Moreover,
we also study reconfiguration of independent sets in three different models and
analyze relationships between these models, observing that shortest path
reconfiguration is a special case of independent set reconfiguration in perfect
graphs, under any of the three models. Finally, we give polynomial results for
restricted classes of graphs (even-hole-free and -free graphs)
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
Reconfiguring k-Path Vertex Covers
A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The K-PATH VERTEX COVER RECONFIGURATION (K-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-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 K-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k=2, known as the VERTEX COVER RECONFIGURATION (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes can be extended for K-PVCR. In particular, we prove a complexity dichotomy for K-PVCR on general graphs: on those whose maximum degree is three (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is two (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for K-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest
Token Jumping in minor-closed classes
Given two -independent sets and of a graph , one can ask if it
is possible to transform the one into the other in such a way that, at any
step, we replace one vertex of the current independent set by another while
keeping the property of being independent. Deciding this problem, known as the
Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar
graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by
if the input graph is -free.
We prove that the result of Ito et al. can be extended to any
-free graphs. In other words, if is a -free
graph, then it is possible to decide in FPT-time if can be transformed into
. As a by product, the TJ-reconfiguration problem is FPT in many well-known
classes of graphs such as any minor-free class