A remark on the (2,2)-domination number

• Autorzy: Torsten Korneffel , Dirk Meierling , Lutz Volkmann
• Języki publikacji: EN

Abstrakty

EN

A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter $γ_{k,p}(G)$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that $γ_{k,p}(G) ≤ (p/(p+k))n(G)$ for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that $γ_{2,2}(G) ≤ (n(G)+1)/2$ for all connected graphs G and characterize all connected graphs with $γ_{2,2} = (n+1)/2$. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ₂ ≥ 3.

Słowa kluczowe

EN

domination; distance domination number; p-domination number

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2008
• Tom: 28
• Numer: 2
• Strony: 361-366

Daty

wydano

2008

otrzymano

2007-05-02

poprawiono

2008-03-25

zaakceptowano

2008-03-25

Twórcy

autor

Torsten Korneffel

• Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

autor

Dirk Meierling

• Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

autor

Lutz Volkmann

• Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

Bibliografia

• [1] T.J. Bean, M.A. Henning and H.C. Swart, On the integrity of distance domination in graphs, Australas. J. Combin. 10 (1994) 29-43.
• [2] E.J. Cockayne, B. Gamble and B. Shepherd, An upper bound for the k-domination number of a graph, J. Graph Theory 9 (1985) 101-102, doi: 10.1002/jgt.3190090414.
• [3] M. Fischermann and L. Volkmann, A remark on a conjecture for the (k,p)-domination number, Util. Math. 67 (2005) 223-227.
• [4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
• [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998).
• [6] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975) 225-233.

Identyfikatory

DOI

10.7151/dmgt.1411