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2013 | 33 | 1 | 139-146
Tytuł artykułu

On Graphs with Disjoint Dominating and 2-Dominating Sets

Treść / Zawartość
Warianty tytułu
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
Wydawca
Rocznik
Tom
33
Numer
1
Strony
139-146
Opis fizyczny
Daty
wydano
2013-03-01
online
2013-04-13
Twórcy
  • Department of Mathematics University of Johannesburg Auckland Park, 2006 South Africa, mahenning@uj.ac.za
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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1652
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