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Tytuł:
Signed domination and signed domatic numbers of digraphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743935.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
oriented graph
signed dominating function
signed domination number
signed domatic number
Opis:
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.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 3; 415-427
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Weak signed Roman k-domination in digraphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/29519480.pdf
Data publikacji:
2024
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
digraph
weak signed Roman k-dominating function
weak signed Roman k-domination number
signed Roman k-dominating function
signed Roman k-domination number
Opis:
Let $ k ≥ 1 $ be an integer, and let $ D $ be a finite and simple digraph with vertex set $ V (D) $. A weak signed Roman k-dominating function (WSRkDF) on a digraph $ D $ is a function $ f : V (D) → {−1, 1, 2} $ satisfying the condition that $ \Sigma_{x∈N^−[v]} f(x) ≥ k $ for each v ∈ V (D), where $ N^− [v] $ consists of $ v $ and all vertices of $ D $ from which arcs go into $ v $. The weight of a WSRkDF $ f $ is $ w(f) = \Sigma_{v∈V} (D) f(v) $. The weak signed Roman k-domination number $ \gamma_{wsR}^k (D) $ is the minimum weight of a WSRkDF on $ D $. In this paper we initiate the study of the weak signed Roman k-domination number of digraphs, and we present different bounds on $ \gamma_{wsR}^k (D) $. In addition, we determine the weak signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number $ \gamma_{wsR} (D) = \gamma_{wsR}^1 (D) $ and the signed Roman k-domination number $ \gamma_{sR}^k (D) $.
Źródło:
Opuscula Mathematica; 2024, 44, 2; 285-296
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds on the Signed 2-Independence Number in Graphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/29794119.pdf
Data publikacji:
2013-09-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
bounds
signed 2-independence function
signed 2-independence number
Nordhaus-Gaddum type result
Opis:
Let $G$ be a finite and simple graph with vertex set $V (G)$, and let $f V (G) → {−1, 1}$ be a two-valued function. If $∑_{x∈N|v|} f(x) ≤ 1$ for each $v ∈ V (G)$, where $N[v]$ is the closed neighborhood of $v$, then $f$ is a signed 2-independence function on $G$. The weight of a signed 2-independence function $f$ is $w(f) = ∑_{v∈V (G)} f(v)$. The maximum of weights $w(f)$, taken over all signed 2-independence functions $f$ on $G$, is the signed 2-independence number $α_s^2(G)$ of $G$. In this work, we mainly present upper bounds on $α_s^2(G)$, as for example $α_s^2(G) ≤ n−2 [∆ (G)//2]$, and we prove the Nordhaus-Gaddum type inequality $α_s^2 (G) + α_s^2(G) ≤ n+1$, where $n$ is the order and $∆ (G)$ is the maximum degree of the graph $G$. Some of our theorems improve well-known results on the signed 2-independence number.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 4; 709-715
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Signed Total Roman k-Domatic Number Of A Graph
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31341581.pdf
Data publikacji:
2017-11-27
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
signed total Roman k-dominating function
signed total Roman k-domination number
signed total Roman k-domatic number
Opis:
Let $ k \ge 1 $ be an integer. A signed total Roman $k$-dominating function on a graph $G$ is a function $ f : V (G) \rightarrow {−1, 1, 2} $ such that $ \Sigma_{ u \in N(v) } f(u) \ge k $ for every $ v \in V (G) $, where $ N(v) $ is the neighborhood of $ v $, and every vertex $ u \in V (G) $ for which $ f(u) = −1 $ is adjacent to at least one vertex w for which $ f(w) = 2 $. A set $ { f_1, f_2, . . ., f_d} $ of distinct signed total Roman $k$-dominating functions on $G$ with the property that $ \Sigma_{i=1}^d f_i(v) \le k $ for each $ v \in V (G) $, is called a signed total Roman $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed total Roman $k$-dominating family on $G$ is the signed total Roman $k$-domatic number of $G$, denoted by $ d_{stR}^k (G) $. In this paper we initiate the study of signed total Roman $k$-domatic numbers in graphs, and we present sharp bounds for $ d_{stR}^k (G) $. In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman $k$-domatic number of some graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 4; 1027-1038
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Sufficient Conditions for Maximally Edge-Connected and Super-Edge-Connected Graphs Depending on The Clique Number
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31343389.pdf
Data publikacji:
2019-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
edge-connectivity
clique number
maximally edge-connected graphs
super-edge-connected graphs
Opis:
Let G be a connected graph with minimum degree δ and edge-connectivity λ. A graph is maximally edge-connected if λ = δ, and it is super-edgeconnected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. The clique number ω(G) of a graph G is the maximum cardinality of a complete subgraph of G. In this paper, we show that a connected graph G with clique number ω(G) ≤ r is maximally edge-connected or super-edge-connected if the number of edges is large enough. These are generalizations of corresponding results for triangle-free graphs by Volkmann and Hong in 2017.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 2; 567-573
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Double Roman Domatic Number of a Digraph
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31348166.pdf
Data publikacji:
2020-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
double Roman domination
double Roman domatic number
Opis:
A double Roman dominating function on a digraph $D$ with vertex set $V(D)$ is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function $f : V (D) → {0, 1, 2, 3}$ having the property that if $f(v) = 0$, then the vertex $v$ must have at least two in-neighbors assigned 2 under $f$ or one in-neighbor w with $f(w) = 3$, and $if f(v) = 1$, then the vertex v must have at least one in-neighbor $u$ with $f(u) ≥ 2$. A set ${f_1, f_2, . . ., f_d}$ of distinct double Roman dominating functions on $D$ with the property that $∑_{i=1}^df_i(v)≤3$ for each $v ∈ V (D)$ is called a double Roman dominating family (of functions) on $D$. The maximum number of functions in a double Roman dominating family on $D$ is the double Roman domatic number of $D$, denoted by $d_{dR}(D)$. We initiate the study of the double Roman domatic number, and we present different sharp bounds on $d_{dR}(D)$. In addition, we determine the double Roman domatic number of some classes of digraphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 4; 995-1004
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Connected global offensive k-alliances in graphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743583.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
alliances in graphs
connected global offensive k-alliance
global offensive k-alliance
domination
Opis:
We consider finite graphs G with vertex set V(G). For a subset S ⊆ V(G), we define by G[S] the subgraph induced by S. By n(G) = |V(G) | and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V(G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N(v) ∩ S | ≥ |N(v) -S | + k for every vertex v ∈ V(G) -S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number $γₒ^{k,c}(G)$ is the minimum cardinality of a connected global offensive k-alliance in G.
In this paper we characterize connected graphs G with $γₒ^{k,c}(G) = n(G)$. In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with $γₒ^{k,c}(G) = n(G) - 1$. Furthermore, we present different tight bounds of $γₒ^{k,c}(G)$.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 4; 699-707
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Upper Bounds on the Signed Total (k, k)-Domatic Number of Graphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31339301.pdf
Data publikacji:
2015-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
signed total (k
k)-domatic number
signed total k-dominating function
signed total k-domination number
regular graphs
Opis:
Let $G$ be a graph with vertex set $V (G)$, and let $ f : V (G) \rightarrow {−1, 1}$ be a two-valued function. If $ k \geq 1$ is an integer and \( \sum_{ x \in N(v)} f(x) \geq k \) for each $ v \in V (G) $, where $N(v)$ is the neighborhood of $v$, then $f$ is a signed total $k$-dominating function on $G$. A set ${f_1, f_2, . . ., f_d}$ of distinct signed total k-dominating functions on $G$ with the property that \( \sum_{i=1}^d f_i(x) \leq k \) for each $ x \in V (G)$, is called a signed total ($k$, $k$)-dominating family (of functions) on $G$. The maximum number of functions in a signed total ($k$, $k$)-dominating family on $G$ is the signed total ($k$, $k$)-domatic number of $G$. In this article we mainly present upper bounds on the signed total ($k$, $k$)- domatic number, in particular for regular graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 4; 641-650
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Signed Total Roman Domination in Digraphs
Autorzy:
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31342127.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
signed total Roman dominating function
signed total Roman domination number
Opis:
Let $D$ be a finite and simple digraph with vertex set $V (D)$. A signed total Roman dominating function (STRDF) on a digraph $D$ is a function $ f : V (D) \rightarrow {−1, 1, 2} $ satisfying the conditions that (i) $ \Sigma_{x \in N^− (v) } f(x) \ge 1 $ for each $ v \in V (D) $, where $ N^− (v) $ consists of all vertices of $D$ from which arcs go into $v$, and (ii) every vertex u for which $f(u) = −1$ has an inner neighbor $v$ for which $f(v) = 2$. The weight of an STRDF $f$ is $ w(f) = \Sigma_{ v \in V } (D) f(v) $. The signed total Roman domination number $ \gamma_{stR} (D) $ of $D$ is the minimum weight of an STRDF on $D$. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on $ \gamma_{stR} (D) $. In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number $ \gamma_{stR} (G)$ of graphs $G$.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 261-272
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Characterization of trees with equal 2-domination number and domination number plus two
Autorzy:
Chellali, Mustapha
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743587.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
2-domination number
domination number
trees
Opis:
Let G = (V(G),E(G)) be a simple graph, and let k be a positive integer. A subset D of V(G) is a k-dominating set if every vertex of V(G) - D is dominated at least k times by D. The k-domination number γₖ(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ₂(T) ≥ γ₁(T)+1 and characterized extremal trees attaining this bound. In this paper we characterize all trees T with γ₂(T) = γ₁(T)+2.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 4; 687-697
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Roman bondage in graphs
Autorzy:
Rad, Nader
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743601.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
Roman domination
Roman bondage number
Opis:
A Roman dominating function on a graph G is a function f:V(G) → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f(V(G)) = ∑_{u ∈ V(G)}f(u)$. The Roman domination number, $γ_R(G)$, of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_R(G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E' ⊆ E(G) for which $γ_R(G -E') > γ_R(G)$. We determine the Roman bondage number in several classes of graphs and give some sharp bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 4; 763-773
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A lower bound for the irredundance number of trees
Autorzy:
Poschen, Michael
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743933.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
irredundance
tree
domination
Opis:
Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound
γ(T) ≥ (n(T) + 2 - n₁(T))/3.
In this paper we prove
ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 2; 209-215
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The total {k}-domatic number of digraphs
Autorzy:
Sheikholeslami, Seyed
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743233.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
digraph
total {k}-dominating function
total {k}-domination number
total {k}-domatic number
Opis:
For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V(D) to the set {0,1,2, ...,k} such that for any vertex v ∈ V(D), the condition $∑_{u ∈ N^{ -}(v)}f(u) ≥ k$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set ${f₁,f₂, ...,f_d}$ of total {k}-dominating functions of D with the property that $∑_{i = 1}^d f_i(v) ≤ k$ for each v ∈ V(D), is called a total {k}-dominating family (of functions) on D. The maximum number of functions in a total {k}-dominating family on D is the total {k}-domatic number of D, denoted by $dₜ^{{k}}(D)$. Note that $dₜ^{{1}}(D)$ is the classic total domatic number $dₜ(D)$. In this paper we initiate the study of the total {k}-domatic number in digraphs, and we present some bounds for $dₜ^{{k}}(D)$. Some of our results are extensions of well-know properties of the total domatic number of digraphs and the total {k}-domatic number of graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 3; 461-471
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Characterization of block graphs with equal 2-domination number and domination number plus one
Autorzy:
Hansberg, Adriana
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743677.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
2-domination
multiple domination
block graph
Opis:
Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G).
If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 1; 93-103
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The k-rainbow domatic number of a graph
Autorzy:
Sheikholeslami, Seyyed
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743715.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
k-rainbow dominating function
k-rainbow domination number
k-rainbow domatic number
Opis:
For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2, ...,k} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃_{u ∈ N(v)}f(u) = {1,2, ...,k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set ${f₁,f₂, ...,f_d}$ of k-rainbow dominating functions on G with the property that $∑_{i = 1}^d |f_i(v)| ≤ k$ for each v ∈ V(G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by $d_{rk}(G)$. Note that $d_{r1}(G)$ is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for $d_{rk}(G)$. Many of the known bounds of d(G) are immediate consequences of our results.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 1; 129-140
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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