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Wyświetlanie 1-15 z 122
Tytuł:
On the (2,2)-domination number of trees
Autorzy:
Lu, You
Hou, Xinmin
Xu, Jun-Ming
Powiązania:
https://bibliotekanauki.pl/articles/744559.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
total domination number
(2,2)-domination number
Opis:
Let γ(G) and $γ_{2,2}(G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2(γ(T)+1))/3 ≤ γ_{2,2}(T) ≤ 2γ(T)$. Moreover, we characterize all the trees achieving the equalities.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 2; 185-199
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:
Weak roman domination in graphs
Autorzy:
Roushini Leely Pushpam, P.
Malini Mai, T.
Powiązania:
https://bibliotekanauki.pl/articles/743835.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
weak Roman domination number
Opis:
Let G = (V,E) be a graph and f be a function f:V → {0,1,2}. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f': V → {0,1,2} defined by f'(u) = 1, f'(v) = f(v)-1 and f'(w) = f(w) if w ∈ V-{u,v}, has no undefended vertex. The weight of f is $w(f) = ∑_{v ∈ V}f(v)$. The weak Roman domination number, denoted by $γ_r(G)$, is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which $γ_r(G) = γ(G)$ and find $γ_r$-value for a caterpillar, a 2×n grid graph and a complete binary tree.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 1; 161-170
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:
Bounds on the Locating Roman Domination Number in Trees
Autorzy:
Jafari Rad, Nader
Rahbani, Hadi
Powiązania:
https://bibliotekanauki.pl/articles/16647912.pdf
Data publikacji:
2018-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Roman domination number
locating domination number
locating Roman domination number
tree
Opis:
A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → {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 an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = {v ∈ V : f(v) = i} for i = 0, 1, 2. An RDF f = (V0, V1, V2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V2 ≠ N(v) ∩ V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number $\gamma _R^L (G)$ is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 1; 49-62
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds on the Locating Roman Domination Number in Trees
Autorzy:
Jafari Rad, Nader
Rahbani, Hadi
Powiązania:
https://bibliotekanauki.pl/articles/31342446.pdf
Data publikacji:
2018-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Roman domination number
locating domination number
locating Roman domination number
tree
Opis:
A Roman dominating function (or just RDF) on a graph $ G = (V, E) $ is a function $ f : V \rightarrow \{ 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 an RDF $f$ is the value $ f(V (G)) = \Sigma_{ u \in V (G) } f(u) $. An RDF $f$ can be represented as $ f = (V_0, V_1, V_2) $, where $ V_i = \{ v \in V : f(v) = i \} $ for $ i = 0, 1, 2 $. An RDF $ f = (V_0, V_1, V_2) $ is called a locating Roman dominating function (or just LRDF) if $ N(u) \cap V_2 \ne N(v) \cap V_2 $ for any pair $u$, $v$ of distinct vertices of $ V_0 $. The locating Roman domination number $ \gamma_R^L (G) $ is the minimum weight of an LRDF of $G$. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 1; 49-62
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Gallai-type equality for the total domination number of a graph
Autorzy:
Zhou, Sanming
Powiązania:
https://bibliotekanauki.pl/articles/744259.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
total domination number
Gallai equality
Opis:
We prove the following Gallai-type equality
γₜ(G) + εₜ(G) = p
for any graph G with no isolated vertex, where p is the number of vertices of G, γₜ(G) is the total domination number of G, and εₜ(G) is the maximum integer s such that there exists a spanning forest F with s the number of pendant edges of F minus the number of star components of F.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 3; 539-543
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On locating-domination in graphs
Autorzy:
Chellali, Mustapha
Mimouni, Malika
Slater, Peter
Powiązania:
https://bibliotekanauki.pl/articles/744569.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
upper locating-domination number
locating-domination number
Opis:
A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number $γ_L(G)$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, $Γ_L(G)$ is the maximum cardinality of a minimal LDS of G. We present different bounds on $Γ_L(G)$ and $γ_L(G)$.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 2; 223-235
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Accurate Domination in Graphs
Autorzy:
Cyman, Joanna
Henning, Michael A.
Topp, Jerzy
Powiązania:
https://bibliotekanauki.pl/articles/31343372.pdf
Data publikacji:
2019-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
accurate domination number
tree
corona
Opis:
A dominating set of a graph G is a subset D ⊆ VG such that every vertex not in D is adjacent to at least one vertex in D. The cardinality of a smallest dominating set of G, denoted by γ(G), is the domination number of G. The accurate domination number of G, denoted by γa(G), is the cardinality of a smallest set D that is a dominating set of G and no |D|-element subset of VG \ D is a dominating set of G. We study graphs for which the accurate domination number is equal to the domination number. In particular, all trees G for which γa(G) = γ(G) are characterized. Furthermore, we compare the accurate domination number with the domination number of different coronas of a graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 3; 615-627
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Domination parameters of a graph with added vertex
Autorzy:
Zwierzchowski, M.
Powiązania:
https://bibliotekanauki.pl/articles/2050876.pdf
Data publikacji:
2004
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
total domination number
strong domination number
subdivision
Opis:
Let $G = (V, E)$ be a graph. A subset $D \subseteq V$ is a total dominating set of $G$ if for every vertex $y \in V$ there is a vertex $x \in D$ with $xy \in E$. A subset $D \subseteq V$ is a strong dominating set of G if for every vertex $y \in V - D$ there is a vertex $x \in D$ with $xy \in E and deg_{G}(x) \geq deg_{G}(y)$. The total domination number $\gamma_{t}(G)$ (the strong domination number $\gamma_{S}(G)$) is defined as the minimum cardinality of a total dominating set (a strong dominating set) of $G$. The concept of total domination was first defined by Cockayne, Dawes and Hedetniemi in 1980 [1], while the strong domination was introduced by Sampathkumar and Pushpa Latha in 1996 [3]. By a subdivision of an edge $uv \in E$ we mean removing edge $uv$, adding a new vertex $x$, and adding edges $ux$ and $vx$. A graph obtained from $G$ by subdivision an edge $uv \in E$ is denoted by $G \oplus uxvx$. The behaviour of the total domination number and the strong domination number of a graph $G \oplus u_{x}v_{x}$ is developed.
Źródło:
Opuscula Mathematica; 2004, 24, 2; 231-234
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total outer-connected domination in trees
Autorzy:
Cyman, Joanna
Powiązania:
https://bibliotekanauki.pl/articles/744028.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total outer-connected domination number
domination number
Opis:
Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by $γ_{tc}(G)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then $γ_{tc}(T) ≥ ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 3; 377-383
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Independent Domination in Planar Cubic Graphs
Autorzy:
Abrishami, Gholamreza
Henning, Michael A.
Rahbarnia, Freydoon
Powiązania:
https://bibliotekanauki.pl/articles/31343229.pdf
Data publikacji:
2019-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
independent domination number
domination number
cubic graphs
Opis:
A set $S$ of vertices in a graph $G$ is an independent dominating set of $G$ if $S$ is an independent set and every vertex not in $S$ is adjacent to a vertex in $S$. The independent domination number, $i(G)$, of $G$ is the minimum cardinality of an independent dominating set. Goddard and Henning [Discrete Math. 313 (2013) 839–854] posed the conjecture that if \( G \not\in \{ K_{3,3}, C_5 \square K_2 \} \) is a connected, cubic graph on $n$ vertices, then $i(G) \le 3/8 n $, where $ C_5 \square K_2 $ is the 5-prism. As an application of known result, we observe that this conjecture is true when $G$ is 2-connected and planar, and we provide an infinite family of such graphs that achieve the bound. We conjecture that if $G$ is a bipartite, planar, cubic graph of order $n$, then $ i(G) \le 1/3 n $, and we provide an infinite family of such graphs that achieve this bound.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 4; 841-853
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Domination Number, Independent Domination Number and 2-Independence Number in Trees
Autorzy:
Dehgardi, Nasrin
Sheikholeslami, Seyed Mahmoud
Valinavaz, Mina
Aram, Hamideh
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/32083746.pdf
Data publikacji:
2021-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
2-independence number
domination number
independent domination number
Opis:
For a graph $G$, let $\gamma(G)$ be the domination number, $i(G)$ be the independent domination number and $\beta_2(G)$ be the 2-independence number. In this paper, we prove that for any tree $T$ of order $n ≥ 2, 4\beta_2(T) − 3\gamma(T) ≥ 3i(T)$, and we characterize all trees attaining equality. Also we prove that for every tree $T$ of order \(n ≥ 2, i(T)≤\frac{3\beta_2(T)}{4}\), and we characterize all extreme trees.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 1; 39-49
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Trees with equal restrained domination and total restrained domination numbers
Autorzy:
Raczek, Joanna
Powiązania:
https://bibliotekanauki.pl/articles/743684.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total restrained domination number
restrained domination number
trees
Opis:
For a graph G = (V,E), a set D ⊆ V(G) is a total restrained dominating set if it is a dominating set and both ⟨D⟩ and ⟨V(G)-D⟩ do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V(G) is a restrained dominating set if it is a dominating set and ⟨V(G)-D⟩ does not contain an isolated vertex. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We characterize all trees for which total restrained and restrained domination numbers are equal.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 1; 83-91
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Slater and Sub-k-Domination Number of a Graph with Applications to Domination and k-Domination
Autorzy:
Amos, David
Asplund, John
Brimkov, Boris
Davila, Randy
Powiązania:
https://bibliotekanauki.pl/articles/32083827.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Slater number
domination number
sub- k -domination number
k -domination number
degree sequence index strategy
Opis:
In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted subk(G). This invariant serves as a generalization of the Slater number; in particular, we show that subk(G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination numbers of several families of graphs, provide structural results on sub-k-domination, and explore properties of graphs which are subk(G)-critical with respect to addition and deletion of vertices and edges.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 209-225
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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