104 research outputs found
The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs
We study the problem of transforming one list (vertex) coloring of a graph
into another list coloring by changing only one vertex color assignment at a
time, while at all times maintaining a list coloring, given a list of allowed
colors for each vertex. This problem is known to be PSPACE-complete for
bipartite planar graphs. In this paper, we first show that the problem remains
PSPACE-complete even for bipartite series-parallel graphs, which form a proper
subclass of bipartite planar graphs. We note that our reduction indeed shows
the PSPACE-completeness for graphs with pathwidth two, and it can be extended
for threshold graphs. In contrast, we give a polynomial-time algorithm to solve
the problem for graphs with pathwidth one. Thus, this paper gives precise
analyses of the problem with respect to pathwidth
Using contracted solution graphs for solving reconfiguration problems.
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Using Contracted Solution Graphs for Solving Reconfiguration Problems
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Using contracted solution graphs for solving reconfiguration problems
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Coloring Reconfiguration Problems and Their Generalizations
Tohoku University周暁課
Reconfiguring Graph Colorings
Graph coloring has been studied for a long time and continues to receive
interest within the research community \cite{kubale2004graph}. It has applications
in scheduling \cite{daniel2004graph}, timetables, and compiler register
allocation \cite{lewis2015guide}. The most popular variant of graph coloring,
k-coloring, can be thought of as an assignment of colors to the vertices of a
graph such that adjacent vertices are assigned different colors.
Reconfiguration problems, typically defined on the solution space of search problems,
broadly ask whether one solution can be transformed to another solution using
step-by-step transformations, when constrained to one or more specific transformation
steps \cite{van2013complexity}. One well-studied reconfiguration problem is the
problem of deciding whether one k-coloring can be transformed to another k-coloring
by changing the color of one vertex at a time, while always maintaining a k-coloring
at each step.
We consider two variants of graph coloring: acyclic coloring and equitable
coloring, and their corresponding reconfiguration problems. A k-acylic coloring is
a k-coloring where there are more than two colors used by the vertices of each
cycle, and a k-equitable coloring is a k-coloring such that each color class, which is
defined as the set of all vertices with a particular color, is nearly the same
size as all others.
We show that reconfiguration of acyclic colorings is PSPACE-hard, and that for
non-bipartite graphs with chromatic number 3 there exist two k-acylic colorings
and such that there is no sequence of transformations that can
transform to . We also consider the problem of whether two
k-acylic colorings can be transformed to each other in at most steps, and
show that it is in XP, which is the class of algorithms that run in time
for some computable function and parameter , where in this
case the parameter is defined to be the length of the reconfiguration sequence
plus the length of the longest induced cycle.
We also show that the reconfiguration of equitable colorings is PSPACE-hard
and W[1]-hard with respect to the number of vertices with the same color. We
give polynomial-time algorithms for Reconfiguration of Equitable Colorings when
the number of colors used is two and also for paths when the number of colors
used is three
Counting subgraphs of coloring graphs
The chromatic polynomial of a graph can be viewed as
counting the number of vertices in a family of coloring graphs associated with (proper) -colorings of as a function of the
number of colors . These coloring graphs can be understood as a
reconfiguration system. We generalize the chromatic polynomial to
, counting occurrences of arbitrary induced subgraphs in
these coloring graphs, and we prove that these functions are polynomial in .
In particular, we study the chromatic pairs polynomial ,
which counts the number of edges in coloring graphs, corresponding to the
number of pairs of colorings that differ on a single vertex. We show two trees
share a chromatic pairs polynomial if and only if they have the same degree
sequence, and we conjecture that the chromatic pairs polynomial refines the
chromatic polynomial in general. We also instantiate our polynomials with other
choices of to generate new graph invariants.Comment: 25 pages, 14 figure
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