An interpolation theorem for partitions which are complete with respect to hereditary properties

Cockayne, E. J.; Miller, Gary Glenn; Prins, Geert

doi:10.1016/0095-8956(72)90065-2

Cited by 18 publications

(4 citation statements)

References 5 publications

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“…Considering the above problems we indirectly use interpolation result of Cockayne et al (1972) who proved that for every hereditary class P and every graph G the number of colors used by the greedy algorithm can take any value from χ P (G) to Γ P (G). We shall also use a fact that both χ P (G) and Γ P (G) are monotone with respect to taking induced subgraphs, i.e., if H ≤ G, then χ P (H ) ≤ χ P (G) and Γ P (H ) ≤ Γ P (G).…”

Section: Grundy P-coloringmentioning

confidence: 99%

On Computational Aspects of Greedy Partitioning of Graphs

Borowiecki

2017

Frontiers in Algorithmics

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In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related too greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs P-coloring with at least k colors is NPcomplete if P is a class of K p -free graphs with p ≥ 3. On the other hand we give a polynomial-time algorithm when k is fixed and the family of minimal forbidden graphs defining the class P is finite. We also prove coNP-completeness of deciding if for a given graph G and an integer t ≥ 0 the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any P-coloring of G is bounded by t. In view of computational hardness, we present new Brooks-type bound on the largest number of colors used by the greedy P-coloring algorithm.

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Section: Grundy P-coloringmentioning

confidence: 99%

On Computational Aspects of Greedy Partitioning of Graphs

Borowiecki

2017

Frontiers in Algorithmics

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“…x, PI, and y . A partial generalization was provided by Cockayne, Miller, and Prins [4]. Recall that a property P is hereditary if each induced subgraph of a graph G with property P also enjoys that property.…”

Section: Several Interpolating Invariants Have Already Been Presentedmentioning

confidence: 99%

Maximum versus minimum invariants for graphs

Harary¹

1983

Journal of Graph Theory

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Two examples in graph theory of the concept of the maximum and minimum pair of invariants resulting from a graphical parameter are ( I ) the chromatic number and its maximum version, the achromatic number, and (2) the genus and the maximum genus. The primary purpose of this expository survey is to propose the generalization of this idea to each invariant p of a maximum or minimum nature. As illustrations, we consider for p the number of independent points or lines, the point arboricity, the connectivity, the rectilinear crossing number, the diameter of a spanning tree, and the conventional extremal number with respect to a forbidden subgraph. An additional objective is to describe the idea of an "interpolating invariant" which takes on, in a precise sense, all values between its minimum and its maximum, and to give examples.

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“…In improper colourings (sometimes called generalized, defective or relaxed) adjacent vertices may be assigned the same colour, but some other constraint is placed on colour classes (see e.g. [1,2,5,6,7,8,9,11] and [14]).…”

Section: Introductionmentioning

confidence: 99%

On generating sets of induced-hereditary properties

Semanišin¹

2002

Discuss. Math. Graph Theory

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A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization Theorem for induced-hereditary properties of graphs.

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An interpolation theorem for partitions which are complete with respect to hereditary properties

Cited by 18 publications

References 5 publications

On Computational Aspects of Greedy Partitioning of Graphs

On Computational Aspects of Greedy Partitioning of Graphs

Maximum versus minimum invariants for graphs

On generating sets of induced-hereditary properties

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