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

• Volume: 19, Issue: 1, page 71-78
• ISSN: 2083-5892

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## Abstract

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Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G\left[{V}_{i}\right]$ has the property ${}_{i}$. A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.

## How to cite

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Jozef Bucko, and Jaroslav Ivančo. "Uniquely partitionable planar graphs with respect to properties having a forbidden tree." Discussiones Mathematicae Graph Theory 19.1 (1999): 71-78. <http://eudml.org/doc/270406>.

@article{JozefBucko1999,
abstract = {Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G[V_i]$ has the property $_i$. A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.},
author = {Jozef Bucko, Jaroslav Ivančo},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {uniquely partitionable planar graphs; forbidden graphs; partition; planar graphs; hereditary additive properties; forbidden tree},
language = {eng},
number = {1},
pages = {71-78},
title = {Uniquely partitionable planar graphs with respect to properties having a forbidden tree},
url = {http://eudml.org/doc/270406},
volume = {19},
year = {1999},
}

TY - JOUR
AU - Jozef Bucko
AU - Jaroslav Ivančo
TI - Uniquely partitionable planar graphs with respect to properties having a forbidden tree
JO - Discussiones Mathematicae Graph Theory
PY - 1999
VL - 19
IS - 1
SP - 71
EP - 78
AB - Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G[V_i]$ has the property $_i$. A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.
LA - eng
KW - uniquely partitionable planar graphs; forbidden graphs; partition; planar graphs; hereditary additive properties; forbidden tree
UR - http://eudml.org/doc/270406
ER -

## References

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1. [1] J. Bucko, M. Frick, P. Mihók and R. Vasky, Uniquely partitionable graphs, Discuss. Math. Graph Theory 17 (1997) 103-114, doi: 10.7151/dmgt.1043. Zbl0906.05057
2. [2] J. Bucko, P. Mihók and M. Voigt, Uniquely partitionable planar graphs, Discrete Math. 191 (1998) 149-158, doi: 10.1016/S0012-365X(98)00102-2. Zbl0957.05029
3. [3] M. Borowiecki, J. Bucko, P. Mihók, Z. Tuza and M. Voigt, Remarks on the existence of uniquely partitionable planar graphs, 13. Workshop on Discrete Optimization, Burg, abstract, 1998.
4. [4] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985) 49-58. Zbl0623.05043

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