Structural Parameterizations for Equitable Coloring

Gomes, Guilherme C. M.; Guedes, Matheus R.; dos, Santos, Vinicius F.

doi:10.48550/arxiv.1911.03297

Cited by 2 publications

(10 citation statements)

References 14 publications

(24 reference statements)

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“…It is known that ECP does not admit a polynomial kernel when parameterized by vertex cover and the number of colors unless NP ⊆ coNP/poly [18]. Hence the same result is true for ECP when parameterized by vertex cover alone.…”

Section: Budgeted Graph Coloring Parameterized By the Vertex Cover Sizementioning

confidence: 78%

“…It follows from Corollary 2 that BCP is NP-hard on interval graphs and co-graphs as ECP is NP-hard on these class of graphs [18]. Similarly as ECP is W [1]-hard on bounded treewidth graphs [12], BCP is also W [1]-hard parameterized by treewidth.…”

Section: Budgeted Coloring Problem (Bcp)mentioning

confidence: 91%

“…Even though the classical coloring problem parameterized by CVD number is folklore FPT result, for the completeness of our paper we briefly describe an approach to solve the problem in FPT time. Due to [18], it is known that ECP is FPT parameterized by CVD. Next, by showing a parameter preserving reduction to ECP we prove that the coloring problem is FPT when parameterized by CVD.…”

Section: -Partition Problemmentioning

confidence: 99%

“…BCP is trivial on cliques. Though BCP can be solved in polynomial time on cluster graphs by constructing flow network [18], we give an alternate simpler algorithm. Next, we have the following lemma.…”

Section: On Cluster Graphsmentioning

confidence: 99%

“…Without loss of generality assume that, in α for each i, P i is colored with color i. Though the following lemma follows form Lemma 1 and Lemma 2 in [18], we give a slightly different proof just for completeness. Lemma 6.…”

Section: Bcp Parameterized By Cluster Vertex Deletion Number (Cvd)mentioning

confidence: 99%

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Structural Parameterizations of Budgeted Graph Coloring

Bandopadhyay¹,

Banerjee²,

Banik³

et al. 2021

Preprint

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We introduce a variant of the graph coloring problem, which we denote as Budgeted Coloring Problem (BCP). Given a graph G, an integer c and an ordered list of integers {b 1 , b 2 , . . . , b c }, BCP asks whether there exists a proper coloring of G where the i-th color is used to color at most b i many vertices. This problem generalizes two well-studied graph coloring problems, Bounded Coloring Problem (BoCP) and Equitable Coloring Problem (ECP) and as in the case of other coloring problems, it is NP-hard even for constant values of c. So we study BCP under the paradigm of parameterized complexity, particularly with respect to (structural) parameters that specify how far (the deletion distance) the input graph is from a tractable graph class.• We show that BCP is FPT (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for ECP and immediately extends to the BoCP, which was earlier not known. • We show that BCP is polynomial time solvable for cluster graphs generalizing a similar result for ECP. However, we show that BCP is FPT, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for ECP for the same parameter. • While the BoCP is known to be polynomial time solvable on split graphs, we show that BCP is NP-hard on split graphs. As BoCP is hard on bipartite graphs when c > 3, the result follows for BCP as well. We provide a dichotomy result by showing that BCP is polynomial time solvable on bipartite graphs when c = 2. We also show that BCP is NP-hard on co-cluster graphs, contrasting the polynomial time algorithm for ECP and BoCP. Finally we present an O * (2 |V (G)| ) algorithm for the BCP, generalizing the known algorithm with a similar bound for the standard chromatic number.

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Section: Budgeted Graph Coloring Parameterized By the Vertex Cover Sizementioning

confidence: 78%

Section: Budgeted Coloring Problem (Bcp)mentioning

confidence: 91%

Section: -Partition Problemmentioning

confidence: 99%

Section: On Cluster Graphsmentioning

confidence: 99%

Section: Bcp Parameterized By Cluster Vertex Deletion Number (Cvd)mentioning

confidence: 99%

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Structural Parameterizations of Budgeted Graph Coloring

Bandopadhyay¹,

Banerjee²,

Banik³

et al. 2021

Preprint

View full text Add to dashboard Cite

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Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

Furmańczyk¹,

Mkrtchyan²

2020

Preprint

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An equitable coloring of a graph G = (V, E) is a (proper) vertex-coloring of G, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.

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Structural Parameterizations for Equitable Coloring

Cited by 2 publications

References 14 publications

Structural Parameterizations of Budgeted Graph Coloring

Structural Parameterizations of Budgeted Graph Coloring

Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

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