# Graphs with convex domination number close to their order

• Volume: 26, Issue: 2, page 307-316
• ISSN: 2083-5892

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## Abstract

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For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance ${d}_{G}\left(u,v\right)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length ${d}_{G}\left(u,v\right)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number ${\gamma }_{con}\left(G\right)$ of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.

## How to cite

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Joanna Cyman, Magdalena Lemańska, and Joanna Raczek. "Graphs with convex domination number close to their order." Discussiones Mathematicae Graph Theory 26.2 (2006): 307-316. <http://eudml.org/doc/270504>.

@article{JoannaCyman2006,
abstract = {For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance $d_G(u,v)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length $d_G(u,v)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number $γ_\{con\}(G)$ of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.},
author = {Joanna Cyman, Magdalena Lemańska, Joanna Raczek},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {convex domination; Cartesian product},
language = {eng},
number = {2},
pages = {307-316},
title = {Graphs with convex domination number close to their order},
url = {http://eudml.org/doc/270504},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Joanna Cyman
AU - Magdalena Lemańska
AU - Joanna Raczek
TI - Graphs with convex domination number close to their order
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 2
SP - 307
EP - 316
AB - For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance $d_G(u,v)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length $d_G(u,v)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number $γ_{con}(G)$ of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.
LA - eng
KW - convex domination; Cartesian product
UR - http://eudml.org/doc/270504
ER -

## References

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1. [1] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs (Marcel Dekker, Inc., 1998). Zbl0890.05002
2. [2] Sergio R. Canoy Jr and I.J.L. Garces, Convex sets under some graphs operations, Graphs and Combinatorics 18 (2002) 787-793, doi: 10.1007/s003730200065. Zbl1009.05054

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