125 research outputs found

    Linear Transformations Between Colorings in Chordal Graphs

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    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

    Recoloring graphs of treewidth 2

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    Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any (d+2)(d + 2)-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for d=1d = 1. Bousquet and Perarnau proved that a linear transformation exists for between (2d+2)(2d + 2)-colorings. It is open to determine if this bound can be reduced. In this note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist

    Data Reduction for Graph Coloring Problems

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    This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

    Distributed Recoloring of Interval and Chordal Graphs

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    Rainbow domination and related problems on some classes of perfect graphs

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    Let k∈Nk \in \mathbb{N} and let GG be a graph. A function f:V(G)→2[k]f: V(G) \rightarrow 2^{[k]} is a rainbow function if, for every vertex xx with f(x)=∅f(x)=\emptyset, f(N(x))=[k]f(N(x)) =[k]. The rainbow domination number γkr(G)\gamma_{kr}(G) is the minimum of ∑x∈V(G)∣f(x)∣\sum_{x \in V(G)} |f(x)| over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs
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