307 research outputs found
Recoloring bounded treewidth graphs
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Any graph is
-mixing, where is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two -colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
Jerrum proved that any graph is -mixing if is at least the maximum
degree plus two. We improve Jerrum's bound using the grundy number, which is
the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure
The Complexity of Rerouting Shortest Paths
The Shortest Path Reconfiguration problem has as input a graph G (with unit
edge lengths) with vertices s and t, and two shortest st-paths P and Q. The
question is whether there exists a sequence of shortest st-paths that starts
with P and ends with Q, such that subsequent paths differ in only one vertex.
This is called a rerouting sequence.
This problem is shown to be PSPACE-complete. For claw-free graphs and chordal
graphs, it is shown that the problem can be solved in polynomial time, and that
shortest rerouting sequences have linear length. For these classes, it is also
shown that deciding whether a rerouting sequence exists between all pairs of
shortest st-paths can be done in polynomial time. Finally, a polynomial time
algorithm for counting the number of isolated paths is given.Comment: The results on claw-free graphs, chordal graphs and isolated paths
have been added in version 2 (april 2012). Version 1 (September 2010) only
contained the PSPACE-hardness result. (Version 2 has been submitted.
Recoloring graphs via tree decompositions
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Jerrum proved that any
graph is -mixing if is at least the maximum degree plus two. We first
improve Jerrum's bound using the grundy number, which is the worst number of
colors in a greedy coloring.
Any graph is -mixing, where is the treewidth of the graph
(Cereceda 2006). We prove that the shortest sequence between any two
-colorings is at most quadratic (which is optimal up to a constant
factor), a problem left open in Bonamy et al. (2012).
We also prove that given any two -colorings of a cograph (resp.
distance-hereditary graph) , we can find a linear (resp. quadratic) sequence
between them. In both cases, the bounds cannot be improved by more than a
constant factor for a fixed . The graph classes are also optimal in
some sense: one of the smallest interesting superclass of distance-hereditary
graphs corresponds to comparability graphs, for which no such property holds
(even when relaxing the constraint on the length of the sequence). As for
cographs, they are equivalently the graphs with no induced , and there
exist -free graphs that admit no sequence between two of their
-colorings.
All the proofs are constructivist and lead to polynomial-time recoloring
algorithmComment: 20 pages, 8 figures, partial results already presented in
http://arxiv.org/abs/1302.348
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
Linear Transformations Between Colorings in Chordal Graphs
Let k and d be such that k >= d+2. Consider two k-colorings of a d-degenerate graph G. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length.
If k=d+2, we know that there exists graphs for which a quadratic number of recolorings is needed. And when k=2d+2, there always exists a linear transformation. In this paper, we prove that, as long as k >= d+4, there exists a transformation of length at most f(Delta) * n between any pair of k-colorings of chordal graphs (where Delta denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two k-colorings c_1,c_2 computes a linear transformation between c_1 and c_2
Minimum Sum Edge Colorings of Multicycles
In the minimum sum edge coloring problem, we aim to assign natural numbers to
edges of a graph, so that adjacent edges receive different numbers, and the sum
of the numbers assigned to the edges is minimum. The {\em chromatic edge
strength} of a graph is the minimum number of colors required in a minimum sum
edge coloring of this graph. We study the case of multicycles, defined as
cycles with parallel edges, and give a closed-form expression for the chromatic
edge strength of a multicycle, thereby extending a theorem due to Berge. It is
shown that the minimum sum can be achieved with a number of colors equal to the
chromatic index. We also propose simple algorithms for finding a minimum sum
edge coloring of a multicycle. Finally, these results are generalized to a
large family of minimum cost coloring problems
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
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