# On Graphs with Disjoint Dominating and 2-Dominating Sets

Michael A. Henning; Douglas F. Rall

Discussiones Mathematicae Graph Theory (2013)

- Volume: 33, Issue: 1, page 139-146
- ISSN: 2083-5892

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topMichael A. Henning, and Douglas F. Rall. "On Graphs with Disjoint Dominating and 2-Dominating Sets." Discussiones Mathematicae Graph Theory 33.1 (2013): 139-146. <http://eudml.org/doc/268032>.

@article{MichaelA2013,

abstract = {A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the graph.},

author = {Michael A. Henning, Douglas F. Rall},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {domination; 2-domination; vertex partition},

language = {eng},

number = {1},

pages = {139-146},

title = {On Graphs with Disjoint Dominating and 2-Dominating Sets},

url = {http://eudml.org/doc/268032},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Michael A. Henning

AU - Douglas F. Rall

TI - On Graphs with Disjoint Dominating and 2-Dominating Sets

JO - Discussiones Mathematicae Graph Theory

PY - 2013

VL - 33

IS - 1

SP - 139

EP - 146

AB - A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the graph.

LA - eng

KW - domination; 2-domination; vertex partition

UR - http://eudml.org/doc/268032

ER -

## References

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- [10] J. Southey and M.A. Henning, Dominating and total dominating partitions in cubic graphs, Central European J. Math. 9(3) (2011) 699-708. doi:10.7151/s11533-011-0014-2[Crossref] Zbl1233.05151
- [11] J. Southey and M.A. Henning, A characterization of graphs with disjoint dominating and paired-dominating sets, J. Comb. Optim. 22 (2011) 217-234. doi:10.1007/s10878-009-9274-1[Crossref][WoS] Zbl1232.05172
- [12] B. Zelinka, Total domatic number and degrees of vertices of a graph, Math. Slovaca 39 (1989) 7-11. Zbl0688.05066

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