2,387 research outputs found
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
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
Independent-Set Reconfiguration Thresholds of Hereditary Graph Classes
Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly k vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most k fewer vertices than the initial solution. We are interested in determining the minimum value of k for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large
Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited
Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + ?, where ? is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if ? is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case where ? is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3
Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited
Given a graph and two vertex sets satisfying a certain feasibility condition,
a reconfiguration problem asks whether we can reach one vertex set from the
other by repeating prescribed modification steps while maintaining feasibility.
In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic
meta-theorem for reconfiguration problems that says if the feasibility can be
expressed in monadic second-order logic (MSO), then the problem is
fixed-parameter tractable parameterized by , where
is the number of steps allowed to reach the target set. On the other
hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if is not
part of the parameter, then the problem is PSPACE-complete even on graphs of
bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case
where is not part of the parameter, using some structural graph
parameters incomparable with bandwidth. We show that if the feasibility is
defined in MSO, then the reconfiguration problem under the so-called token
jumping rule is fixed-parameter tractable parameterized by neighborhood
diversity. We also show that the problem is fixed-parameter tractable
parameterized by , where is the size of sets being
transformed. We finally complement the positive result for treedepth by showing
that the problem is PSPACE-complete on forests of depth .Comment: 25 pages, 2 figures, ESA 202
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