# On arbitrarily vertex decomposable unicyclic graphs with dominating cycle

• Volume: 26, Issue: 3, page 403-412
• ISSN: 2083-5892

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## Abstract

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A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that ${\sum }_{i=1}^{k}{n}_{i}=n$, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set ${V}_{i}$ induces a connected subgraph of G on ${n}_{i}$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.

## How to cite

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Sylwia Cichacz, and Irmina A. Zioło. "On arbitrarily vertex decomposable unicyclic graphs with dominating cycle." Discussiones Mathematicae Graph Theory 26.3 (2006): 403-412. <http://eudml.org/doc/270726>.

@article{SylwiaCichacz2006,
abstract = {A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that $∑^k_\{i=1\} n_i = n$, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set $V_i$ induces a connected subgraph of G on $n_i$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.},
author = {Sylwia Cichacz, Irmina A. Zioło},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {arbitrarily vertex decomposable graph; dominating cycle},
language = {eng},
number = {3},
pages = {403-412},
title = {On arbitrarily vertex decomposable unicyclic graphs with dominating cycle},
url = {http://eudml.org/doc/270726},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Sylwia Cichacz
AU - Irmina A. Zioło
TI - On arbitrarily vertex decomposable unicyclic graphs with dominating cycle
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 3
SP - 403
EP - 412
AB - A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that $∑^k_{i=1} n_i = n$, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set $V_i$ induces a connected subgraph of G on $n_i$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.
LA - eng
KW - arbitrarily vertex decomposable graph; dominating cycle
UR - http://eudml.org/doc/270726
ER -

## References

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1. [1] D. Barth, O. Baudon and J. Puech, Decomposable trees: a polynomial algorithm for tripodes, Discrete Appl. Math. 119 (2002) 205-216, doi: 10.1016/S0166-218X(00)00322-X. Zbl1002.68107
2. [2] D. Barth and H. Fournier, A degree bound on decomposable trees, Discrete Math. 306 (2006) 469-477, doi: 10.1016/j.disc.2006.01.006. Zbl1092.05054
3. [3] S. Cichacz, A. Görlich, A. Marczyk, J. Przybyło and M. Woźniak, Arbitrarily vertex decomposable caterpillars with four or five leaves, Preprint MD-010 (2005), http://www.ii.uj.edu.pl/preMD/, to appear. Zbl1142.05065
4. [4] M. Hornák and M. Woźniak, Arbitrarily vertex decomposable trees are of maximum degree at most six, Opuscula Math. 23 (2003) 49-62. Zbl1093.05510
5. [5] R. Kalinowski, M. Pilśniak, M. Woźniak and I.A. Zioło, Arbitrarily vertex decomposable suns with few rays, preprint (2005), http://www.ii.uj.edu.pl/preMD/. Zbl1214.05125
6. [6] A. Marczyk, Ore-type condition for arbitrarily vertex decomposable graphs, preprint (2005).

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