On Graphs with Disjoint Dominating and 2-Dominating Sets

• Autorzy: Michael A. Henning , Douglas F. Rall
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

domination; 2-domination; vertex partition

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 1
• Strony: 139-146

Daty

wydano

2013-03-01

online

2013-04-13

Twórcy

autor

Michael A. Henning

• Department of Mathematics University of Johannesburg Auckland Park, 2006 South Africa

autor

Douglas F. Rall

• Department of Mathematics Furman University Greenville, SC, USA

Bibliografia

• [1] M. Dorfling, W. Goddard, M.A. Henning and C.M. Mynhardt, Construction of trees and graphs with equal domination parameters, DiscreteMath. 306 (2006) 2647-2654. doi:10.1016/j.disc.2006.04.031[Crossref]
• [2] S.M. Hedetniemi, S.T. Hedetniemi, R.C. Laskar, L. Markus and P.J. Slater, Disjoint dominating sets in graphs, Proc. ICDM 2006, Ramanujan Mathematics Society Lecture Notes Series 7 (2008) 87-100.
• [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• [4] M.A. Henning, C. Löwenstein and D. Rautenbach, Remarks about disjoint dominating sets, Discrete Math. 309 (2009) 6451-6458. doi:10.1016/j.disc.2009.06.017[WoS][Crossref]
• [5] M.A. Henning, C. Löwenstein and D. Rautenbach, An independent dominating set in the complement of a minimum dominating set of a tree, Appl. Math. Lett. 23 (2010) 79-81. doi:10.1016/j.aml.2009.08.008[Crossref][WoS]
• [6] M.A. Henning, C. Löwenstein and D. Rautenbach, Partitioning a graph into a dominating set, a total dominating set, and something else, Discuss. Math. Graph Theory 30 (2010) 563-574. doi:10.7151/dmgt.1514[Crossref]
• [7] M.A. Henning and J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162.
• [8] M.A. Henning and J. Southey, A characterization of graphs with disjoint dominating and total dominating sets, Quaest. Math. 32 (2009) 119-129.[WoS][Crossref]
• [9] O. Ore, Theory of Graphs: Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc., Providence, RI, 1962).
• [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]
• [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]
• [12] B. Zelinka, Total domatic number and degrees of vertices of a graph, Math. Slovaca 39 (1989) 7-11.

Identyfikatory

DOI

10.7151/dmgt.1652