# Domination Subdivision Numbers

Teresa W. Haynes; Sandra M. Hedetniemi; Stephen T. Hedetniemi; David P. Jacobs; James Knisely; Lucas C. van der Merwe

Discussiones Mathematicae Graph Theory (2001)

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

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topTeresa 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

top- [1] Arumugam, private communication, June 2000.
- [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] D. Hare and W. McCuaig, A characterization of graphs whose domination and matching numbers are equal, unpublished manuscript, 1998.
- [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] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
- [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] T.W. Haynes, S.T. Hedetniemi and L.C. van der Merwe, Total domination subdivision numbers, submitted. Zbl1020.05048
- [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] 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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