73 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
The complexity of dominating set reconfiguration
Suppose that we are given two dominating sets and of a graph
whose cardinalities are at most a given threshold . Then, we are asked
whether there exists a sequence of dominating sets of between and
such that each dominating set in the sequence is of cardinality at most
and can be obtained from the previous one by either adding or deleting
exactly one vertex. This problem is known to be PSPACE-complete in general. In
this paper, we study the complexity of this decision problem from the viewpoint
of graph classes. We first prove that the problem remains PSPACE-complete even
for planar graphs, bounded bandwidth graphs, split graphs, and bipartite
graphs. We then give a general scheme to construct linear-time algorithms and
show that the problem can be solved in linear time for cographs, trees, and
interval graphs. Furthermore, for these tractable cases, we can obtain a
desired sequence such that the number of additions and deletions is bounded by
, where is the number of vertices in the input graph
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
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
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees
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
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