Connected Greedy Colourings

Benevides, Fabrício Siqueira; Campos, Víctor; Dourado, Mitre C.; Griffiths, Simon; Morris, Robert; Sampaio, Leonardo; Silva, Ana

doi:10.1007/978-3-642-54423-1_38

Cited by 6 publications

(25 citation statements)

References 9 publications

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“…Benevides, Campos, Dourado, Griffiths, Morris, Sampaio and Silva [2] have recently proven that χ c (G) cannot be arbitrarily large with respect to χ(G). The difference can be at most 1:…”

Section: Introductionmentioning

confidence: 99%

“…In general, χ c (G) is not equal to χ(G); see [2,Theorem 2]. We similarly define the connected Grundy number of a graph G, denoted Γ c (G), as the maximum number of colours for connected orderings:…”

Section: Introductionmentioning

confidence: 99%

“…1 All other graphs are called bad. 2 A graph G for which no connected ordering achieves the optimal value χ(G), i.e. χ c (G) > χ(G), is called ugly.…”

Section: Introductionmentioning

confidence: 99%

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Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

Beaudou,

Brosse,

Defrain

et al. 2021

Preprint

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The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering σ = v1, . . . , vn, the "First-Fit" greedy colouring algorithm colours the vertices in the order of σ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain connected greedy colourings. For some graphs, all connected greedy colourings use exactly χ(G) colours; they are called good graphs. On the opposite, some graphs do not admit any connected greedy colouring using only χ(G) colours; they are called ugly graphs. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no K4-minor free graph is ugly.

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“…Benevides, Campos, Dourado, Griffiths, Morris, Sampaio and Silva [2] have recently proven that χ c (G) cannot be arbitrarily large with respect to χ(G). The difference can be at most 1:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

Beaudou,

Brosse,

Defrain

et al. 2021

Preprint

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show abstract

“…Connected Grundy Coloring was introduced by Benevides et al [3], who proved it to be NP-complete, even for chordal graphs and for co-bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

“…While for the parameter "number of colors", Grundy Coloring is in XP and Weak Grundy Coloring is in FPT, we show that Connected Grundy Coloring is NP-complete even when k = 7, that is, it does not belong to XP unless P = NP. Note that the known NP-hardness proof of [3] for Connected Grundy Coloring was only for an unbounded number of colors.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Grundy Coloring and Its Variants

Bonnet

Foucaud

Kim

et al. 2015

Lecture Notes in Computer Science

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International audienceThe Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We show that Grundy Coloring can be solved in time O∗(2.443n) on graphs of order n. While the problem is known to be solvable in time f(k,w)⋅n for graphs of treewidth w, we prove that under the Exponential Time Hypothesis, it cannot be computed in time O∗(cw), for any constant c. We also study the parameterized complexity of Grundy Coloring parameterized by the number of colors, showing that it is in FPTfor graphs including chordal graphs, claw-free graphs, and graphs excluding a fixed minor.Finally, we consider two previously studied variants of Grundy Coloring, namely Weak Grundy Coloring and Connected Grundy Coloring. We show that Weak Grundy Coloring is fixed-parameter tractable with respect to the weak Grundy number. In stark contrast, it turns out that checking whether a given graph has connected Grundy number at least k is NP-complete already for k=7

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