22 research outputs found
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
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
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
Reconfiguration of Cliques in a Graph
We study reconfiguration problems for cliques in a graph, which determine
whether there exists a sequence of cliques that transforms a given clique into
another one in a step-by-step fashion. As one step of a transformation, we
consider three different types of rules, which are defined and studied in
reconfiguration problems for independent sets. We first prove that all the
three rules are equivalent in cliques. We then show that the problems are
PSPACE-complete for perfect graphs, while we give polynomial-time algorithms
for several classes of graphs, such as even-hole-free graphs and cographs. In
particular, the shortest variant, which computes the shortest length of a
desired sequence, can be solved in polynomial time for chordal graphs,
bipartite graphs, planar graphs, and bounded treewidth graphs
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
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
Shortest Reconfiguration of Matchings
Imagine that unlabelled tokens are placed on the edges of a graph, such that
no two tokens are placed on incident edges. A token can jump to another edge if
the edges having tokens remain independent. We study the problem of determining
the distance between two token configurations (resp., the corresponding
matchings), which is given by the length of a shortest transformation. We give
a polynomial-time algorithm for the case that at least one of the two
configurations is not inclusion-wise maximal and show that otherwise, the
problem admits no polynomial-time sublogarithmic-factor approximation unless P
= NP. Furthermore, we show that the distance of two configurations in bipartite
graphs is fixed-parameter tractable parameterized by the size of the
symmetric difference of the source and target configurations, and obtain a
-factor approximation algorithm for every if
additionally the configurations correspond to maximum matchings. Our two main
technical tools are the Edmonds-Gallai decomposition and a close relation to
the Directed Steiner Tree problem. Using the former, we also characterize those
graphs whose corresponding configuration graphs are connected. Finally, we show
that deciding if the distance between two configurations is equal to a given
number is complete for the class , and deciding if the diameter of
the graph of configurations is equal to is -hard.Comment: 31 pages, 3 figure
The Galactic Center Black Hole Laboratory
The super-massive 4 million solar mass black hole Sagittarius~A* (SgrA*)
shows flare emission from the millimeter to the X-ray domain. A detailed
analysis of the infrared light curves allows us to address the accretion
phenomenon in a statistical way. The analysis shows that the near-infrared
flare amplitudes are dominated by a single state power law, with the low states
in SgrA* limited by confusion through the unresolved stellar background. There
are several dusty objects in the immediate vicinity of SgrA*. The source G2/DSO
is one of them. Its nature is unclear. It may be comparable to similar stellar
dusty sources in the region or may consist predominantly of gas and dust. In
this case a particularly enhanced accretion activity onto SgrA* may be expected
in the near future. Here the interpretation of recent data and ongoing
observations are discussed.Comment: 30 pages - 7 figures - accepted for publication by Springer's
"Fundamental Theories of Physics" series; summarizing GC contributions of 2
conferences: 'Equations of Motion in Relativistic Gravity' at the
Physikzentrum Bad Honnef, Bad Honnef, Germany, (Feb. 17-23, 2013) and the
COST MP0905 'The Galactic Center Black Hole Laboratory' Granada, Spain (Nov.
19 - 22, 2013
Vertex Cover Reconfiguration and Beyond
Abstract. In the Vertex Cover Reconfiguration (VCR) problem, given graph G = (V,E), positive integers k and `, and two vertex cov-ers S and T of G of size at most k, we determine whether S can be transformed into T by a sequence of at most ` vertex additions or re-movals such that each operation results in a vertex cover of size at most k. Motivated by recent results establishing the W[1]-hardness of VCR when parameterized by `, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR re-mains W[1]-hard on bipartite graphs, is NP-hard but fixed-parameter tractable on graphs of bounded degree, and is solvable in time polyno-mial in |V (G) | on even-hole-free graphs and cactus graphs. We prove W[1]-hardness and fixed-parameter tractability via two new problems of independent interest.