Domination numbers in graphs with removed edge or set of edges

• Volume: 25, Issue: 1-2, page 51-56
• ISSN: 2083-5892

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Abstract

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It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number ${\gamma }_{w}$ and the connected domination number ${\gamma }_{c}$, i.e., we show that ${\gamma }_{w}\left(G\right)\le {\gamma }_{w}\left(G-e\right)\le {\gamma }_{w}\left(G\right)+1$ and ${\gamma }_{c}\left(G\right)\le {\gamma }_{c}\left(G-e\right)\le {\gamma }_{c}\left(G\right)+2$ if G and G-e are connected. Additionally we show that ${\gamma }_{w}\left(G\right)\le {\gamma }_{w}\left(G-Eₚ\right)\le {\gamma }_{w}\left(G\right)+p-1$ and ${\gamma }_{c}\left(G\right)\le {\gamma }_{c}\left(G-Eₚ\right)\le {\gamma }_{c}\left(G\right)+2p-2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order p is a connected subgraph of G.

How to cite

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Magdalena Lemańska. "Domination numbers in graphs with removed edge or set of edges." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 51-56. <http://eudml.org/doc/270684>.

@article{MagdalenaLemańska2005,
abstract = {It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_w$ and the connected domination number $γ_c$, i.e., we show that $γ_w(G) ≤ γ_w(G-e) ≤ γ_w(G)+1$ and $γ_c(G) ≤ γ_c(G-e) ≤ γ_c(G) + 2$ if G and G-e are connected. Additionally we show that $γ_w(G) ≤ γ_w(G-Eₚ) ≤ γ_w(G) + p - 1$ and $γ_c(G) ≤ γ_c(G -Eₚ) ≤ γ_c(G) + 2p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order p is a connected subgraph of G.},
author = {Magdalena Lemańska},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {connected domination number; weakly connected domination number; edge removal},
language = {eng},
number = {1-2},
pages = {51-56},
title = {Domination numbers in graphs with removed edge or set of edges},
url = {http://eudml.org/doc/270684},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Magdalena Lemańska
TI - Domination numbers in graphs with removed edge or set of edges
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 1-2
SP - 51
EP - 56
AB - It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_w$ and the connected domination number $γ_c$, i.e., we show that $γ_w(G) ≤ γ_w(G-e) ≤ γ_w(G)+1$ and $γ_c(G) ≤ γ_c(G-e) ≤ γ_c(G) + 2$ if G and G-e are connected. Additionally we show that $γ_w(G) ≤ γ_w(G-Eₚ) ≤ γ_w(G) + p - 1$ and $γ_c(G) ≤ γ_c(G -Eₚ) ≤ γ_c(G) + 2p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order p is a connected subgraph of G.
LA - eng
KW - connected domination number; weakly connected domination number; edge removal
UR - http://eudml.org/doc/270684
ER -

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