Discussiones Mathematicae Graph Theory 31(3) (2011) 415-427
doi: 10.7151/dmgt.1555

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SIGNED DOMINATION AND SIGNED DOMATIC NUMBERS OF DIGRAPHS

Lutz Volkmann

Lehrstuhl II für Mathematik
RWTH-Aachen University
52056 Aachen, Germany
e-mail: volkm@math2.rwth-aachen.de

Abstract

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 γS(D) of D. A set {f1,f2,…,fd} of signed dominating functions on D with the property that ∑i = 1dfi(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 dS(D).

In this work we show that 4−n ≤ γS(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 γS(D)+dS(D) ≤ n+1 for any digraph D of order n, and we characterize the digraphs D with γS(D)+dS(D) = n+1. Some of our theorems imply well-known results on the signed domination number of graphs.

Keywords: digraph, oriented graph, signed dominating function, signed domination number, signed domatic number.

2010 Mathematics Subject Classification: 05C69.

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Received 29 January 2010
Revised 26 April 2010
Accepted 27 April 2010