Graph Subcolorings: Complexity and Algorithms

Fiala, Jiřı́; Jansen, Klaus; Lê, Van Bang; Seidel, Eike

doi:10.1007/3-540-45477-2_15

Cited by 17 publications

(20 citation statements)

References 12 publications

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“…Many variants and generalizations of basic concepts have been introduced and intensively studied over the years. Since this subject is too wide to be surveyed in a short paper, we mention just a few examples like subcoloring known also as P 3 -free coloring (where P p denotes the chordless path on p vertices), P 4 -free coloring and improper coloring, and we refer to appropriate literature on other variants, e.g., many results on subcoloring can be found in Albertson et al [2], Broere and Mynhardt [8], Fiala et al [16] as well as in work of Gimbel and Hartman [17]. For results on P 4 -free coloring see, e.g., Gimbel and Nešetřil [18] and a paper of Hoàng and Le [23], while for improper coloring we refer the reader to papers of Bermond et al [4], Cowen et al [15] and Havet et al [21].…”

Section: Related Research and Our Resultsmentioning

confidence: 99%

Dynamic F-free Coloring of Graphs

Borowiecki

Sidorowicz

2018

Graphs and Combinatorics

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A problem of graph F-free coloring consists in partitioning the vertex set of a graph such that none of the resulting sets induces a graph containing a fixed graph F as an induced subgraph. In this paper we consider dynamic F-free coloring in which, similarly as in online coloring, the graph to be colored is not known in advance; it is gradually revealed to the coloring algorithm that has to color each vertex upon request as well as handle any vertex recoloring requests. Our main concern is the greedy approach and characterization of graph classes for which it is possible to decide in polynomial time whether for the fixed forbidden graph F and positive integer k the greedy algorithm ever uses more than k colors in dynamic F-free coloring. For various classes of graphs we give such characterizations in terms of a finite number of minimal forbidden graphs thus solving the above-mentioned problem for the so-called F-trees when F is 2-connected, and for classical trees, when F is a path of order 3 (the latter variant is also known as subcoloring or 1-improper coloring).

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Section: Related Research and Our Resultsmentioning

confidence: 99%

Dynamic F-free Coloring of Graphs

Borowiecki

Sidorowicz

2018

Graphs and Combinatorics

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“…So, the recognition of classes of graphs for which it is possible to solve the problem in polynomial time is an interesting research topic. Fiala et al [7] showed that, for a fixed s, recognizing s-subcolorability of graphs with tree decomposition bounded by a constant b can be decided in O(n2 b s b+2 ). Stacho [15] presented a polynomial time algorithm for testing the 2-subcolorability of chordal graphs with time complexity of O(n 3 ) and Stacho [16] proved that it is NP-complete to decide, for a given chordal graph G, whether or not G admits a s-subcoloring, s ≥ 3.…”

Section: Subcoloringmentioning

confidence: 99%

Generalizating path and fan graphs: subcoloring and toughness

Markenzon

Waga

2014

Pesqui. Oper.

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ABSTRACT. Two graph classes are presented; the first one (k-ribbon) generalizes the path graph and the second one (k-fan) generalizes the fan graph. We prove that they are subclasses of chordal graphs and so they share the same structural properties of this class. The solution of two problems are presented: the determination of the subchromatic number and the determination of the toughness. It is shown that the elements of the new classes establish bounds for the toughness of k-path graphs.

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“…By setting F = P 3 we get that the sub-coloring problem on general graphs is NP-hard. Fiala et al [9] showed that F -free coloring is NP-hard even for triangle-free planar graphs with maximum degree 4, while giving polynomial time algorithms for sub-coloring on cographs and graphs of bounded tree-width. Stacho [25] has shown that sub-coloring on chordal graphs is NP-complete.…”

Section: Related Workmentioning

confidence: 99%

“…The smallest k for which a graph has a k-sub-coloring is called the sub-chromatic number of G, and is denoted χ s (G). The sub-coloring problem [1,4,5,9,25] seeks to find a partition of vertices of G into the smallest number of sub-color classes. Clearly, any proper coloring of G is also a sub-coloring, since any proper color class can be viewed as the disjoint union of size-1 cliques; hence, χ s (G)≤ χ(G).…”

Section: Introductionmentioning

confidence: 99%

Sub-coloring and Hypo-coloring Interval Graphs

Gandhi

Greening

Pemmaraju

et al. 2010

Graph-Theoretic Concepts in Computer Science

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Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as "subroutines" for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a "forbidden subgraph" characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it.

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Graph Subcolorings: Complexity and Algorithms

Cited by 17 publications

References 12 publications

Dynamic F-free Coloring of Graphs

Dynamic F-free Coloring of Graphs

Generalizating path and fan graphs: subcoloring and toughness

Sub-coloring and Hypo-coloring Interval Graphs

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