24,900 research outputs found
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 Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
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
A reconfigurations analogue of Brooks’ theorem.
Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless
G is a complete graph or a cycle with an odd number of vertices, or
k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike.
We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that
if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter,
if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2).
We complete this structural classification by settling the missing case:
if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2).
We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is
O(n 2) time solvable for k = 3,
PSPACE-complete for 4 ≤ k ≤ Δ(G),
O(n) time solvable for k = Δ(G) + 1,
O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)
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
Reconfiguring Independent Sets in Claw-Free Graphs
We present a polynomial-time algorithm that, given two independent sets in a
claw-free graph , decides whether one can be transformed into the other by a
sequence of elementary steps. Each elementary step is to remove a vertex
from the current independent set and to add a new vertex (not in )
such that the result is again an independent set. We also consider the more
restricted model where and have to be adjacent
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
How the minuscule can contribute to the big picture: the neutron electric dipole moment project at TRIUMF
A permanent electric dipole moment (EDM) of a fundamental particle violates
both parity (P) and time (T) reversal symmetry and combined charge and parity
(CP) reversal symmetry if the combined reversal of charge, parity \textit{and}
time (CPT) is preserved. It is a very promising place to search for physics
beyond the Standard Model. Ultracold neutrons (UCN) are the ideal tool to study
the neutron electric dipole moment since they can be observed for hundreds of
seconds. This article summarizes the current searches for the neutron EDM using
UCN and introduces the project to measure the neutron electric dipole moment at
TRIUMF using its unique accelerator driven spallation neutron and liquid helium
UCN source. The aim is to reach a sensitivity for the neutron EDM of around
cm.Comment: 12 pages, 6 figures, MENU 2016 Conference, Kyoto, Japa
Memory and chaos in an Ising spin glass
The non-equilibrium dynamics of the model 3d-Ising spin glass
- FeMnTiO - has been investigated from the temperature
and time dependence of the zero field cooled magnetization recorded under
certain thermal protocols. The results manifest chaos, rejuvenation and memory
features of the equilibrating spin configuration that are very similar to those
observed in corresponding studies of the archetypal RKKY spin glass Ag(Mn). The
sample is rapidly cooled in zero magnetic field, and the magnetization recorded
on re-heating. When a stop at constant temperature is made during the
cooling, the system evolves toward its equilibrium state at this temperature.
The equilibrated state established during the stop becomes frozen in on further
cooling and is retrieved on re-heating. The memory of the aging at is not
affected by a second stop at a lower temperature
. Reciprocally, the first equilibration at has no influence on
the relaxation at , as expected within the droplet model for domain
growth in a chaotic landscape.Comment: REVTeX style; 4 pages, 4 figure
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