Signed domination and signed domatic numbers of digraphs

• Autorzy: Lutz Volkmann
• Języki publikacji: EN

Abstrakty

EN

Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → {-1,1} be a two-valued function. If $∑_{x ∈ N¯[v]}f(x) ≥ 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number $γ_S(D)$ of D. A set ${f₁,f₂,...,f_d}$ of signed dominating functions on D with the property that $∑_{i = 1}^d f_i(x) ≤ 1$ for each x ∈ V(D), is called a signed dominating family (of functions) on D. The maximum number of functions in a signed dominating family on D is the signed domatic number of D, denoted by $d_S(D)$. In this work we show that $4-n ≤ γ_S(D) ≤ n$ for each digraph D of order n ≥ 2, and we characterize the digraphs attending the lower bound as well as the upper bound. Furthermore, we prove that $γ_S(D) + d_S(D) ≤ n + 1$ for any digraph D of order n, and we characterize the digraphs D with $γ_S(D) + d_S(D) = n + 1$. Some of our theorems imply well-known results on the signed domination number of graphs.

Słowa kluczowe

EN

digraph; oriented graph; signed dominating function; signed domination number; signed domatic number

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2011
• Tom: 31
• Numer: 3
• Strony: 415-427

Daty

wydano

2011

otrzymano

2010-01-29

poprawiono

2010-04-26

zaakceptowano

2010-04-27

Twórcy

autor

Lutz Volkmann

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

Bibliografia

• [1] J.E. Dunbar, S.T. Hedetniemi, M.A. Henning and P.J. Slater, Signed domination in graphs, Graph Theory, Combinatorics, and Applications, John Wiley and Sons, Inc. 1 (1995) 311-322.
• [2] M.A. Henning and P.J. Slater, Inequalities relating domination parameters in cubic graphs, Discrete Math. 158 (1996) 87-98, doi: 10.1016/0012-365X(96)00025-8.
• [3] H. Karami, S.M. Sheikholeslami and A. Khodkar, Lower bounds on the signed domination numbers of directed graphs, Discrete Math. 309 (2009) 2567-2570, doi: 10.1016/j.disc.2008.04.001.
• [4] M. Sheikholeslami and L. Volkmann, Signed domatic number of directed graphs, submitted.
• [5] L. Volkmann and B. Zelinka, Signed domatic number of a graph, Discrete Appl. Math. 150 (2005) 261-267, doi: 10.1016/j.dam.2004.08.010.
• [6] B. Zelinka, Signed domination numbers of directed graphs, Czechoslovak Math. J. 55 (2005) 479-482, doi: 10.1007/s10587-005-0038-5.
• [7] Z. Zhang, B. Xu, Y. Li and L. Liu, A note on the lower bounds of signed domination number of a graph, Discrete Math. 195 (1999) 295-298, doi: 10.1016/S0012-365X(98)00189-7.

Identyfikatory

DOI

10.7151/dmgt.1555