# Signed domination and signed domatic numbers of digraphs

• Volume: 31, Issue: 3, page 415-427
• ISSN: 2083-5892

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## Abstract

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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 ${\sum }_{x\in N¯\left[v\right]}f\left(x\right)\ge 1$ 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 ${\gamma }_{S}\left(D\right)$ of D. A set $f₁,f₂,...,{f}_{d}$ of signed dominating functions on D with the property that ${\sum }_{i=1}^{d}{f}_{i}\left(x\right)\le 1$ 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 ${d}_{S}\left(D\right)$. In this work we show that $4-n\le {\gamma }_{S}\left(D\right)\le n$ 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 ${\gamma }_{S}\left(D\right)+{d}_{S}\left(D\right)\le n+1$ for any digraph D of order n, and we characterize the digraphs D with ${\gamma }_{S}\left(D\right)+{d}_{S}\left(D\right)=n+1$. Some of our theorems imply well-known results on the signed domination number of graphs.

## How to cite

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Lutz Volkmann. "Signed domination and signed domatic numbers of digraphs." Discussiones Mathematicae Graph Theory 31.3 (2011): 415-427. <http://eudml.org/doc/270815>.

@article{LutzVolkmann2011,
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 $γ_S(D)$ 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.},
author = {Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {digraph; oriented graph; signed dominating function; signed domination number; signed domatic number},
language = {eng},
number = {3},
pages = {415-427},
title = {Signed domination and signed domatic numbers of digraphs},
url = {http://eudml.org/doc/270815},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Lutz Volkmann
TI - Signed domination and signed domatic numbers of digraphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 3
SP - 415
EP - 427
AB - 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.
LA - eng
KW - digraph; oriented graph; signed dominating function; signed domination number; signed domatic number
UR - http://eudml.org/doc/270815
ER -

## References

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5. [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. Zbl1079.05071
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