# Uniquely partitionable graphs

• Volume: 17, Issue: 1, page 103-113
• ISSN: 2083-5892

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## Abstract

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G\left[{V}_{i}\right]$ induced by ${V}_{i}$ has property ${}_{i}$; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property , then we say that is additive. A property is called degenerate if there exists a bipartite graph that does not have property . In this paper, we prove that if ₁,..., ₙ are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (₁,...,ₙ)-partitionable graph.

## How to cite

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Jozef Bucko, et al. "Uniquely partitionable graphs." Discussiones Mathematicae Graph Theory 17.1 (1997): 103-113. <http://eudml.org/doc/270374>.

@article{JozefBucko1997,
abstract = {Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G[V_i]$ induced by $V_i$ has property $_i$; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property , then we say that is additive. A property is called degenerate if there exists a bipartite graph that does not have property . In this paper, we prove that if ₁,..., ₙ are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (₁,...,ₙ)-partitionable graph.},
author = {Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hereditary property of graphs; additivity; reducibility; vertex partition; partition; hereditary properties of graphs},
language = {eng},
number = {1},
pages = {103-113},
title = {Uniquely partitionable graphs},
url = {http://eudml.org/doc/270374},
volume = {17},
year = {1997},
}

TY - JOUR
AU - Jozef Bucko
AU - Marietjie Frick
AU - Peter Mihók
TI - Uniquely partitionable graphs
JO - Discussiones Mathematicae Graph Theory
PY - 1997
VL - 17
IS - 1
SP - 103
EP - 113
AB - Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G[V_i]$ induced by $V_i$ has property $_i$; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property , then we say that is additive. A property is called degenerate if there exists a bipartite graph that does not have property . In this paper, we prove that if ₁,..., ₙ are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (₁,...,ₙ)-partitionable graph.
LA - eng
KW - hereditary property of graphs; additivity; reducibility; vertex partition; partition; hereditary properties of graphs
UR - http://eudml.org/doc/270374
ER -

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