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    Wyświetlanie 1-13 z 13
    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:
    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:
    Trees with equal 2-domination and 2-independence numbers
    Autorzy:
    Chellali, Mustapha
    Meddah, Nacéra
    Powiązania:
    https://bibliotekanauki.pl/articles/743338.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    2-domination number
    2-independence number
    trees
    Opis:
    Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V-S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ₂(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β₂(G). Fink and Jacobson proved that γ₂(G) ≤ β₂(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 2; 263-270
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Trees whose 2-domination subdivision number is 2
    Autorzy:
    Atapour, M.
    Sheikholeslami, S. M.
    Khodkar, A.
    Powiązania:
    https://bibliotekanauki.pl/articles/254847.pdf
    Data publikacji:
    2012
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    2-dominating set
    2-domination number
    2-domination subdivision numbe
    Opis:
    A set S of vertices in a graph G = (V,E) is a 2-dominating set if every vertex of V \ S is adjacent to at least two vertices of S. The 2-domination number of a graph G, denoted by γ2(G), is the minimum size of a 2-dominating set of G. The 2-domination subdivision number sdγ2 (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-domination number. The authors have recently proved that for any tree T of order at least 3, 1 ≤ sdγ2 (T ) ≤ 2. In this paper we provide a constructive characterization of the trees whose 2-domination subdivision number is 2.
    Źródło:
    Opuscula Mathematica; 2012, 32, 3; 423-437
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Bounds on the 2-domination number in cactus graphs
    Autorzy:
    Chellali, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/254915.pdf
    Data publikacji:
    2006
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    2-domination number
    total domination number
    independence number
    cactus graphs
    trees
    Opis:
    A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) - k(G), and if G is a graph of order n with at most one cycle, then γ2(G) ≥ (n + l - s)/2 improving Fink and Jacobson's lower bound for trees with l > s, where γt(G), l and s are the total domination number, the number of leaves and support vertices of G, respectively. We also show that if T is a tree of order n ≥ 3, then γ2(T) ≤ β(T) + s - 1, where β(T) is the independence number of T.
    Źródło:
    Opuscula Mathematica; 2006, 26, 1; 5-12
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Criticality indices of 2-rainbow domination of paths and cycles
    Autorzy:
    Bouchou, A.
    Blidia, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/255150.pdf
    Data publikacji:
    2016
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    2-rainbow domination number
    criticality index
    Opis:
    A 2-rainbow dominating function of a graph G (V(G), E(G)) is a function ƒ that assigns to each vertex a set of colors chosen from the set {1,2} so that for each vertex with ƒ (v) = ∅ we have [formula].The weight of a 2RDF ƒ is defined as [formula] minimum weight of a 2RDF is called the 2-rainbow domination number of G, denoted by [formula].The vertex criticality index of a 2-rainbow domination of a graph G is defined as [formula] the edge removal criticality index of a 2-rainbow domination of a graph G is defined as [formula] and the edge addition of a 2-rainbow domination criticality index of G is defined as [formula] where G is the complement graph of G. In this paper, we determine the criticality indices of paths and cycles.
    Źródło:
    Opuscula Mathematica; 2016, 36, 5; 563-574
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Total 2-Rainbow Domination Numbers of Trees
    Autorzy:
    Ahangar, H. Abdollahzadeh
    Amjadi, J.
    Chellali, M.
    Nazari-Moghaddam, S.
    Sheikholeslami, S.M.
    Powiązania:
    https://bibliotekanauki.pl/articles/32083855.pdf
    Data publikacji:
    2021-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    2-rainbow dominating function
    2-rainbow domination number
    total 2-rainbow dominating function
    total 2-rainbow domination number
    Opis:
    A 2-rainbow dominating function (2RDF) of a graph $G = (V(G), E(G))$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set {1, 2} such that for every vertex $v ∈ V(G)$ with $f(v) = ∅$ the condition \(\bigcup_{u∈N(v)}f(u) = \{1, 2\}\) is fulfilled, where $N(v)$ is the open neighborhood of $v$. A total 2-rainbow dominating function $f$ of a graph with no isolated vertices is a 2RDF with the additional condition that the subgraph of $G$ induced by $\{v ∈ V (G) | f(v) ≠∅\}$ has no isolated vertex. The total 2-rainbow domination number, $\gamma_{tr2}(G)$, is the minimum weight of a total 2-rainbow dominating function of $G$. In this paper, we establish some sharp upper and lower bounds on the total 2-rainbow domination number of a tree. Moreover, we show that the decision problem associated with $\gamma_{tr2}(G)$ is NP-complete for bipartite and chordal graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 2; 345-364
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On 2-rainbow domination number of functigraph and its complement
    Autorzy:
    Shaminezhad, Athena
    Vatandoost, Ebrahim
    Powiązania:
    https://bibliotekanauki.pl/articles/255140.pdf
    Data publikacji:
    2020
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    2-rainbow domination number
    functigraph
    complement
    cubic graph
    Opis:
    Let G be a graph and ƒ : V(G) → P({1, 2}) be a function where for every vertex v ∈ V(G), with ƒ (v) = ∅ we have [formula]. Then ƒ is a 2-rainbow dominating function or a 2RDF of G. The weight of ƒ is[formula]. The minimum weight of all 2-rainbow dominating functions is 2-rainbow domination number of G, denoted by [formula]. Let G 1 and G2 be two copies of a graph G with disjoint vertex sets V(G 1) and V(G2), and let σ be a function from V(G 1) to V(G2). We define the functigraph C(G,σ) to be the graph that has the vertex set V(C(G, ,σ)) = V(G 1) U V(G2), and the edge set [formula]. In this paper, 2-rainbow domination number of the functigraph of C(G, ,σ) and its complement are investigated. We obtain a general bound for [formula] and we show that this bound is sharp.
    Źródło:
    Opuscula Mathematica; 2020, 40, 5; 617-627
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    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:
    Roman {2}-Bondage Number of a Graph
    Autorzy:
    Moradi, Ahmad
    Mojdeh, Doost Ali
    Sharifi, Omid
    Powiązania:
    https://bibliotekanauki.pl/articles/32083773.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination
    Roman {2}-domination
    Roman {2}-bondage number
    Opis:
    For a given graph G=(V, E), a Roman {2}-dominating function f : V (G) → {0, 1, 2} has the property that for every vertex u with f(u) = 0, either u is adjacent to a vertex assigned 2 under f, or is adjacent to at least two vertices assigned 1 under f. The Roman {2}-domination number of G, γ{R2}(G), is the minimum of Σu∈V (G) f(u) over all such functions. In this paper, we initiate the study of the problem of finding Roman {2}-bondage number of G. The Roman {2}-bondage number of G, b{R2}, is defined as the cardinality of a smallest edge set E′ ⊆ E for which γ{R2}(G − E′) > γ{R2}(G). We first demonstrate complexity status of the problem by proving that the problem is NP-Hard. Then, we derive useful parametric as well as fixed upper bounds on the Roman {2}-bondage number of G. Specifically, it is known that the Roman bondage number of every planar graph does not exceed 15 (see [S. Akbari, M. Khatirinejad and S. Qajar, A note on the Roman bondage number of planar graphs, Graphs Combin. 29 (2013) 327–331]). We show that same bound will be preserved while computing the Roman {2}-bondage number of such graphs. The paper is then concluded by computing exact value of the parameter for some classes of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 255-268
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Improving some bounds for dominating Cartesian products
    Autorzy:
    Hartnell, Bert
    Rall, Douglas
    Powiązania:
    https://bibliotekanauki.pl/articles/743158.pdf
    Data publikacji:
    2003
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination number
    Cartesian product
    Vizing's conjecture
    2-packing
    Opis:
    The study of domination in Cartesian products has received its main motivation from attempts to settle a conjecture made by V.G. Vizing in 1968. He conjectured that γ(G)γ(H) is a lower bound for the domination number of the Cartesian product of any two graphs G and H. Most of the progress on settling this conjecture has been limited to verifying the conjectured lower bound if one of the graphs has a certain structural property. In addition, a number of authors have established bounds for dominating the Cartesian product of any two graphs. We show how it is possible to improve some of these bounds by imposing conditions on both graphs. For example, we establish a new lower bound for the domination number of T T, when T is a tree, and we improve an upper bound of Vizing in the case when one of the graphs has k > 1 dominating sets which cover the vertex set and the other has a dominating set which partitions in a certain way.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2003, 23, 2; 261-272
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On minimum intersections of certain secondary dominating sets in graphs
    Autorzy:
    Kosiorowska, Anna
    Michalski, Adrian
    Włoch, Iwona
    Powiązania:
    https://bibliotekanauki.pl/articles/29519420.pdf
    Data publikacji:
    2023
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    dominating set
    2-dominating set
    (1, 2)-dominating set
    proper (1, 2)-dominating set
    domination number
    (1,2)-intersection index
    Opis:
    In this paper we consider secondary dominating sets, also named as (1,k)-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the (1, 1)-dominating sets and proper (1, 2)-dominating sets. We introduce (1,2̅)-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.
    Źródło:
    Opuscula Mathematica; 2023, 43, 6; 813-827
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Some properties of the zero divisor graph of a commutative ring
    Autorzy:
    Nazzal, Khalida
    Ghanem, Manal
    Powiązania:
    https://bibliotekanauki.pl/articles/729189.pdf
    Data publikacji:
    2014
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    automorphism group of a graph
    center of a graph
    core of a graph
    k-domination number
    Gaussian integers modulo n
    median of a graph
    2-packing
    perfect graph
    and zero divisor graph
    Opis:
    Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(ℤₙ[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(ℤₙ[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.
    Źródło:
    Discussiones Mathematicae - General Algebra and Applications; 2014, 34, 2; 167-181
    1509-9415
    Pojawia się w:
    Discussiones Mathematicae - General Algebra and Applications
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-13 z 13

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