Criteria of the existence of uniquely partionable graphs with respect to additive induced-hereditary properties

Broere, Izak; Bucko, Jozef; Mihók, Peter

doi:10.7151/dmgt.1156

Cited by 3 publications

(1 citation statement)

References 5 publications

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“…. The binary operation "•" of additive and hereditary properties of finite graphs have been considered in details in [2,3]. For technical reasons we consider also the property Θ = {K 0 } being the smallest graph property in the lattice (M af , ⊆) of all additive inducedhereditary properties of finite character partially ordered by set-inclusion (see [1,14]).…”

Section: Introductionmentioning

confidence: 99%

On infinite uniquely partitionable graphs and graph properties of finite character

Bucko

Mihók

2009

Discuss. Math. Graph Theory

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A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P. Let P 1 , P 2 ,. .. , P n be graph properties of finite character, a graph G is said to be (uniquely) (P 1 , P 2 ,. .. , P n)partitionable if there is an (exactly one) partition {V 1 , V 2 ,. .. , V n } of V (G) such that G[V i ] ∈ P i for i = 1, 2,. .. , n. Let us denote by R = P 1 •P 2 • • • • •P n the class of all (P 1 , P 2 ,. .. , P n)-partitionable graphs. A property R = P 1 •P 2 • • • • •P n , n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P 1 , P 2 ,. .. , P n)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property P i has a weakly universal graph.

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Section: Introductionmentioning

confidence: 99%

On infinite uniquely partitionable graphs and graph properties of finite character

Bucko

Mihók

2009

Discuss. Math. Graph Theory

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Unique Factorization Theorem and Formal Concept Analysis

Mihók

Semanišin

Lecture Notes in Computer Science

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New Results on Generalized Graph Coloring

Alekseev¹,

Farrugia²

2004

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k≥ 3. Recently, this result has been generalized by showing that if all \wp_i's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the \wp_i's are co-additive.

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Criteria of the existence of uniquely partionable graphs with respect to additive induced-hereditary properties

Cited by 3 publications

References 5 publications

On infinite uniquely partitionable graphs and graph properties of finite character

On infinite uniquely partitionable graphs and graph properties of finite character

Unique Factorization Theorem and Formal Concept Analysis

New Results on Generalized Graph Coloring

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