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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:
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
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:
A remark on the (2,2)-domination number
Autorzy:
Korneffel, Torsten
Meierling, Dirk
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743033.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
distance domination number
p-domination number
Opis:
A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter $γ_{k,p}(G)$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that $γ_{k,p}(G) ≤ (p/(p+k))n(G)$ for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that $γ_{2,2}(G) ≤ (n(G)+1)/2$ for all connected graphs G and characterize all connected graphs with $γ_{2,2} = (n+1)/2$. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ₂ ≥ 3.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 361-366
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Extremal bipartite graphs with a unique k-factor
Autorzy:
Hoffmann, Arne
Sidorowicz, Elżbieta
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743924.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
unique k-factor
bipartite graphs
extremal graphs
Opis:
Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size has exactly 2k vertices of degree k and 2k vertices of degree (|V(G)|)/2. As our main result we show that for k ≥ 1, p ≡ t mod k, 0 ≤ t < k, a bipartite graph G of order 2p with a unique k-factor meets 2|E(G)| ≤ p(p+k)-t(k-t). Furthermore, we present the structure of extremal graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 2; 181-192
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
k-independence stable graphs upon edge removal
Autorzy:
Chellali, Mustapha
Haynes, Teresa
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/744261.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
k-independence stable graphs
k-independence
Opis:
Let k be a positive integer and G = (V(G),E(G)) a graph. A subset S of V(G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k-1. The maximum cardinality of a k-independent set of G is the k-independence number βₖ(G). A graph G is called β¯ₖ-stable if βₖ(G-e) = βₖ(G) for every edge e of E(G). First we give a necessary and sufficient condition for β¯ₖ-stable graphs. Then we establish four equivalent conditions for β¯ₖ-stable trees.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 2; 265-274
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the p-domination number of cactus graphs
Autorzy:
Blidia, Mostafa
Chellali, Mustapha
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/744381.pdf
Data publikacji:
2005
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
p-domination number
cactus graphs
Opis:
Let p be a positive integer and G = (V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γₚ(G). It is proved for a cactus graph G that γₚ(G) ⩽ (|V| + |Lₚ(G)| + c(G))/2, for every positive integer p ⩾ 2, where Lₚ(G) is the set of vertices of G of degree at most p-1 and c(G) is the number of odd cycles in G.
Źródło:
Discussiones Mathematicae Graph Theory; 2005, 25, 3; 355-361
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds on the Signed Roman k-Domination Number of a Digraph
Autorzy:
Chen, Xiaodan
Hao, Guoliang
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31343713.pdf
Data publikacji:
2019-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
signed Roman k-dominating function
signed Roman k-domination number
digraph
oriented tree
Opis:
Let $k$ be a positive integer. A signed Roman $k$-dominating function (SRkDF) 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 k $ for each $ v \in V (D) $, where $ N^− [v] $ is the closed in-neighborhood of $v$, and (ii) each vertex $u$ for which $f(u) = −1$ has an in-neighbor $v$ for which $f(v) = 2$. The weight of an SRkDF $f$ is $ \Sigma_{ v \in V (D) } f(v) $. The signed Roman $k$-domination number $ \gamma_{sR}^k (D) $ of a digraph $D$ is the minimum weight of an SRkDF on $D$. We determine the exact values of the signed Roman $k$-domination number of some special classes of digraphs and establish some bounds on the signed Roman $k$-domination number of general digraphs. In particular, for an oriented tree $T$ of order $n$, we show that $ \gamma_{sR}^2 (T) \ge (n + 3)//2 $, and we characterize the oriented trees achieving this lower bound.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 1; 67-79
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the Signed (Total) k-Independence Number in Graphs
Autorzy:
Khodkar, Abdollah
Samadi, Babak
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31234099.pdf
Data publikacji:
2015-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination in graphs
signed k-independence
limited packing
tuple domination
Opis:
Let G be a graph. A function f : V (G) → {−1, 1} is a signed k- independence function if the sum of its function values over any closed neighborhood is at most k − 1, where k ≥ 2. The signed k-independence number of G is the maximum weight of a signed k-independence function of G. Similarly, the signed total k-independence number of G is the maximum weight of a signed total k-independence function of G. In this paper, we present new bounds on these two parameters which improve some existing bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 4; 651-662
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Some remarks on α-domination
Autorzy:
Dahme, Franz
Rautenbach, Dieter
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/744557.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
α-domination
domination
Opis:
Let α ∈ (0,1) and let $G = (V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D ⊆ V_G$ is called an α-dominating set of G, if $|N_G(u) ∩ D| ≥ αd_G(u)$ for all $u ∈ V_G∖D$. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 3; 423-430
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Signed Roman Edge k -Domination in Graphs
Autorzy:
Asgharsharghi, Leila
Sheikholeslami, Seyed Mahmoud
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31342188.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
signed Roman edge k -dominating function
signed Roman edge k -domination number
Opis:
Let $ k \ge 1 $ be an integer, and $ G = (V, E) $ be a finite and simple graph. The closed neighborhood $ N_G [e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and all edges having a common end-vertex with $e$. A signed Roman edge $k$-dominating function (SREkDF) on a graph $G$ is a function $ f : E \rightarrow {−1, 1, 2} $ satisfying the conditions that (i) for every edge $e$ of $G$, $ \Sigma_{ x \in N_G [e] } f(x) \ge k $ and (ii) every edge e for which $f(e) = −1$ is adjacent to at least one edge $ e^′ $ for which $ f(e^′) = 2 $. The minimum of the values $ \Sigma_{e \in E} f(e) $, taken over all signed Roman edge $k$-dominating functions $f$ of $G$ is called the signed Roman edge $k$-domination number of $G$, and is denoted by $ \gamma_{sRk}^' (G) $. In this paper we initiate the study of the signed Roman edge $k$-domination in graphs and present some (sharp) bounds for this parameter.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 39-53
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds on the global offensive k-alliance number in graphs
Autorzy:
Chellali, Mustapha
Haynes, Teresa
Randerath, Bert
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/744476.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
global offensive k-alliance number
independence number
chromatic number
Opis:
Let G = (V(G),E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V(G) is called a global offensive k-alliance if |N(v)∩S| ≥ |N(v)-S|+k for every v ∈ V(G)-S, where N(v) is the neighborhood of v. The global offensive k-alliance number $γₒ^k(G)$ is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on $γₒ^k(G)$ in terms of order, maximum degree, independence number, chromatic number and minimum degree.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 3; 597-613
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The k-Rainbow Bondage Number of a Digraph
Autorzy:
Amjadi, Jafar
Mohammadi, Negar
Sheikholeslami, Seyed Mahmoud
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/31339490.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
k-rainbow dominating function
k-rainbow domination number
k-rainbow bondage number
digraph
Opis:
Let $ D = (V,A) $ be a finite and simple digraph. A $k$-rainbow dominating function ($ k \text{RDF} $) of a digraph $D$ is a function $f$ from the vertex set $V$ to the set of all subsets of the set ${1, 2, . . ., k}$ such that for any vertex $ v \in V $ with $ f(v) = \emptyset $ the condition \( \bigcup_{ u \in N^−(v) } f(u) = {1, 2, . . ., k} \) is fulfilled, where $ N^− (v) $ is the set of in-neighbors of $v$. The weight of a \( k \text{RDF} \) \( f \) is the value \( \omega (f) = \sum_{v \in V} |f(v)| \). The $k$-rainbow domination number of a digraph $D$, denoted by $ \gamma_{rk} (D) $, is the minimum weight of a $ k \text{RDF} $ of $D$. The $k$-rainbow bondage number $ b_{rk} (D) $ of a digraph $D$ with maximum in-degree at least two, is the minimum cardinality of all sets $ A^\prime \subseteq A $ for which $ \gamma_{rk} (D−A^\prime ) > \gamma_{rk} (D) $. In this paper, we establish some bounds for the $k$-rainbow bondage number and determine the $k$-rainbow bondage number of several classes of digraphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 261-270
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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