Uniquely partitionable graphs

Self Cite

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.

Section: Preliminary Resultsmentioning

confidence: 99%

On infinite uniquely partitionable graphs and graph properties of finite character

Bucko

Mihók

2009

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“…This forcing is done as follows. Mihók proved in [2] that for any property P > O not divisible by O, there exists a graph uniquely partitionable into O • P. Take a copy of such a graph and make one vertex from the O part of it adjacent…”

Section: The Np-hardness Resultsmentioning

confidence: 99%

On the computational complexity of (O,P)-partition problems

Kratochvı́l¹,

Schiermeyer²

1997

“…A graph G ∈ P 1 •P 2 is said to be uniquely (P 1 , P 2 )-partitionable if G has exactly one (unordered) vertex (P 1 , P 2 )-partition. For the concept of uniquely partitionable graphs we refer the reader to [1]. Basic properties of uniquely partitionable graphs are discussed in [1] and [4].…”

Section: Introductionmentioning

confidence: 99%

“…For the concept of uniquely partitionable graphs we refer the reader to [1]. Basic properties of uniquely partitionable graphs are discussed in [1] and [4]. The proof used non-planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Bucko¹,

Ivančo²

1999

Self Cite

Let P 1 , P 2 be graph properties. A vertex (P 1 , P 2 )-partition of a graph G is a partition {V 1 , V 2 } of V (G) such that for i = 1, 2 the induced subgraph G[V i ] has the property P i . A property R = P 1 •P 2 is defined to be the set of all graphs having a vertex (P 1 , P 2 )-partition. A graph G ∈ P 1 •P 2 is said to be uniquely (P 1 , P 2 )-partitionable if G has exactly one vertex (P 1 , P 2 )-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.