Signed domination and signed domatic numbers of digraphs

Lutz Volkmann

Discussiones Mathematicae Graph Theory (2011)

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

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 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 f , f , . . . , 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.

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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  4. [4] M. Sheikholeslami and L. Volkmann, Signed domatic number of directed graphs, submitted. Zbl06187634
  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
  6. [6] B. Zelinka, Signed domination numbers of directed graphs, Czechoslovak Math. J. 55 (2005) 479-482, doi: 10.1007/s10587-005-0038-5. Zbl1081.05042
  7. [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. Zbl0928.05052

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