# Domination Subdivision Numbers

• Volume: 21, Issue: 2, page 239-253
• ISSN: 2083-5892

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## Abstract

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A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $s{d}_{\gamma }\left(G\right)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1\le s{d}_{\gamma }\left(G\right)\le 3$ for any graph G. We give a counterexample to this conjecture. On the other hand, we show that $s{d}_{\gamma }\left(G\right)\le \gamma \left(G\right)+1$ for any graph G without isolated vertices, and give constant upper bounds on $s{d}_{\gamma }\left(G\right)$ for several families of graphs.

## How to cite

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Teresa W. Haynes, et al. "Domination Subdivision Numbers." Discussiones Mathematicae Graph Theory 21.2 (2001): 239-253. <http://eudml.org/doc/270326>.

@article{TeresaW2001,
abstract = {A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $sd_γ(G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1 ≤ sd_γ(G) ≤ 3$ for any graph G. We give a counterexample to this conjecture. On the other hand, we show that $sd_γ(G) ≤ γ(G)+1$ for any graph G without isolated vertices, and give constant upper bounds on $sd_γ(G)$ for several families of graphs.},
author = {Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, James Knisely, Lucas C. van der Merwe},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; subdivision},
language = {eng},
number = {2},
pages = {239-253},
title = {Domination Subdivision Numbers},
url = {http://eudml.org/doc/270326},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Teresa W. Haynes
AU - Sandra M. Hedetniemi
AU - Stephen T. Hedetniemi
AU - David P. Jacobs
AU - James Knisely
AU - Lucas C. van der Merwe
TI - Domination Subdivision Numbers
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 2
SP - 239
EP - 253
AB - A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $sd_γ(G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1 ≤ sd_γ(G) ≤ 3$ for any graph G. We give a counterexample to this conjecture. On the other hand, we show that $sd_γ(G) ≤ γ(G)+1$ for any graph G without isolated vertices, and give constant upper bounds on $sd_γ(G)$ for several families of graphs.
LA - eng
KW - domination; subdivision
UR - http://eudml.org/doc/270326
ER -

## References

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1. [1] Arumugam, private communication, June 2000.
2. [2] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079. Zbl0602.05043
3. [3] D. Hare and W. McCuaig, A characterization of graphs whose domination and matching numbers are equal, unpublished manuscript, 1998.
4. [4] T.W. Haynes, S.M. Hedetniemi and S.T. Hedetniemi, Domination and independence subdivision numbers of graphs, Discuss. Math. Graph Theory 20 (2000) 271-280, doi: 10.7151/dmgt.1126. Zbl0984.05066
5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
6. [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, eds, Domination in Graphs (Advanced Topics, Marcel Dekker, Inc., New York, 1998). Zbl0883.00011
7. [7] T.W. Haynes, S.T. Hedetniemi and L.C. van der Merwe, Total domination subdivision numbers, submitted. Zbl1020.05048
8. [8] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104. Zbl0489.05049
9. [9] B. Randerath and L. Volkmann, Characterization of graphs with equal domination and matching number, Utilitas Math. 55 (1999) 65-72. Zbl0940.05058

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