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Wyświetlanie 1-15 z 26
Tytuł:
Domination Parameters of a Graph and its Complement
Autorzy:
Desormeaux, Wyatt J.
Haynes, Teresa W.
Henning, Michael A.
Powiązania:
https://bibliotekanauki.pl/articles/31342430.pdf
Data publikacji:
2018-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
complement
total domination
connected domination
clique domination
restrained domination
Opis:
A dominating set in a graph G is a set S of vertices such that every vertex in V (G) \ S is adjacent to at least one vertex in S, and the domination number of G is the minimum cardinality of a dominating set of G. Placing constraints on a dominating set yields different domination parameters, including total, connected, restrained, and clique domination numbers. In this paper, we study relationships among domination parameters of a graph and its complement.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 1; 203-215
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Domination Versus Paired-Domination in Regular Graphs
Autorzy:
Cyman, Joanna
Dettlaff, Magda
Henning, Michael A.
Lemańska, Magdalena
Raczek, Joanna
Powiązania:
https://bibliotekanauki.pl/articles/31342314.pdf
Data publikacji:
2018-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
total domination
paired-domination
Opis:
A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted γ(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, γt(G), and the paired-domination number, γpr(G), respectively. For k ≥ 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435–447] that γpr(G)/γt(G) ≤ (2k)/(k+1). In the special case when k = 2, we observe that γpr(G)/γt(G) ≤ 4/3, with equality if and only if G ≅ C5. When k = 3, we show that γpr(G)/γt(G) ≤ 3/2, with equality if and only if G is the Petersen graph. More generally for k ≥ 2, if G has girth at least 5 and satisfies γpr(G)/γt(G) = (2k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k ≥ 2 and k ≠ 57, if G has girth at least 5, then γpr(G)/γt(G) ≤ (2k)/(k +1), with equality if and only if k = 2 and G ≅ C5 or k = 3 and G is the Petersen graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 2; 573-586
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Trees with unique minimum total dominating sets
Autorzy:
Haynes, Teresa
Henning, Michael
Powiązania:
https://bibliotekanauki.pl/articles/743354.pdf
Data publikacji:
2002
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
total domination
Opis:
A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.
Źródło:
Discussiones Mathematicae Graph Theory; 2002, 22, 2; 233-246
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Domination in Generalized Prisms and a New Domination Invariant
Autorzy:
Tepeh, Aleksandra
Powiązania:
https://bibliotekanauki.pl/articles/32222717.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
k -rainbow total domination
total domination
Opis:
In this paper we complement recent studies on the total domination of prisms by considering generalized prisms, i.e., Cartesian products of an arbitrary graph and a complete graph. By introducing a new domination invariant on a graph G, called the k-rainbow total domination number and denoted by γkrt(G), it is shown that the problem of finding the total domination number of a generalized prism G □ Kk is equivalent to an optimization problem of assigning subsets of {1, 2, . . ., k} to vertices of G. Various properties of the new domination invariant are presented, including, inter alia, that γkrt(G) = n for a nontrivial graph G of order n as soon as k ≥ 2Δ(G). To prove the mentioned result as well as the closed formulas for the k-rainbow total domination number of paths and cycles for every k, a new weight-redistribution method is introduced, which serves as an efficient tool for establishing a lower bound for a domination invariant.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 1165-1178
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the Complexity of Reinforcement in Graphs
Autorzy:
Rad, Nader Jafari
Powiązania:
https://bibliotekanauki.pl/articles/31340751.pdf
Data publikacji:
2016-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
total domination
total restrained domination
p- domination
k-rainbow domination
reinforcement
NP-hard
Opis:
We show that the decision problem for p-reinforcement, p-total rein- forcement, total restrained reinforcement, and k-rainbow reinforcement are NP-hard for bipartite graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 4; 877-887
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on domination parameters in random graphs
Autorzy:
Bonato, Anthony
Wang, Changping
Powiązania:
https://bibliotekanauki.pl/articles/743019.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
random graphs
independent domination
total domination
Opis:
Domination parameters in random graphs G(n,p), where p is a fixed real number in (0,1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n,p).
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 335-343
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A characterization of (γₜ,γ₂)-trees
Autorzy:
Lu, You
Hou, Xinmin
Xu, Jun-Ming
Li, Ning
Powiązania:
https://bibliotekanauki.pl/articles/744036.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
total domination
2-domination
(λ,μ)-tree
Opis:
Let γₜ(G) and γ₂(G) be the total domination number and the 2-domination number of a graph G, respectively. It has been shown that: γₜ(T) ≤ γ₂(T) for any tree T. In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 3; 425-435
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Protection of Lexicographic Product Graphs
Autorzy:
Klein, Douglas J.
Rodríguez-Velázquez, Juan A.
Powiązania:
https://bibliotekanauki.pl/articles/32361746.pdf
Data publikacji:
2022-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
lexicographic product
weak Roman domination
secure domination
total domination
double total domination
Opis:
In this paper, we study the weak Roman domination number and the secure domination number of lexicographic product graphs. In particular, we show that these two parameters coincide for almost all lexicographic product graphs. Furthermore, we obtain tight bounds and closed formulas for these parameters.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 1; 139-158
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total domination versus paired domination
Autorzy:
Schaudt, Oliver
Powiązania:
https://bibliotekanauki.pl/articles/743224.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
upper total domination
paired domination
upper paired domination
generalized claw-free graphs
Opis:
A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γₚ. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γₚ.
In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γₚ/γₜ ≤ 2 - 2/r for $K_{1,r}$-free graphs. As a consequence, we obtain the sharp bound γₚ/γₜ ≤ 2 - 2/(Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that ${C₅,T_r}$-free graphs fulfill the sharp bound γₚ/γₜ ≤ 2 - 2/r, where $T_r$ is obtained from $K_{1,r}$ by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γₚ/Γₜ. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γₚ ≤ Γₜ holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γₚ/Γₜ taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γₚ/Γₜ ≤ 2 - 2/r for graphs that do not contain the corona of $K_{1,r}$ as subgraph. In particular, we obtain the sharp bound γₚ/Γₜ ≤ 2 - 2/Δ.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 3; 435-447
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbourhood total domination in graphs
Autorzy:
Arumugam, S.
Sivagnanam, C.
Powiązania:
https://bibliotekanauki.pl/articles/254824.pdf
Data publikacji:
2011
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
neighbourhood total domination
total domination
connected domination
paired domination
neighbourhood total domatic number
Opis:
Let G = (V, E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraph ⟨ N(S) ⟩ has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by ϒnt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G). In this paper we initiate a study of these parameters.
Źródło:
Opuscula Mathematica; 2011, 31, 4; 519-531
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Secure domination and secure total domination in graphs
Autorzy:
Klostermeyer, William
Mynhardt, Christina
Powiązania:
https://bibliotekanauki.pl/articles/743322.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
secure domination
total domination
secure total domination
clique covering
Opis:
A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V-X, there exists x ∈ X adjacent to u such that $(X-{x}) ∪ {u}$ is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number $γ_s(G)(γ_{st}(G))$. We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then $γ_{st}(G) ≤ γ_s(G)$. We also show that $γ_{st}(G)$ is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 267-284
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Some results on total domination in direct products of graphs
Autorzy:
Dorbec, Paul
Gravier, Sylvain
Klavžar, Sandi
Spacapan, Simon
Powiązania:
https://bibliotekanauki.pl/articles/743885.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
direct product
total domination
k-tuple domination
open packing
domination
Opis:
Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 1; 103-112
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds On The Disjunctive Total Domination Number Of A Tree
Autorzy:
Henning, Michael A.
Naicker, Viroshan
Powiązania:
https://bibliotekanauki.pl/articles/31341124.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
disjunctive total domination
trees
Opis:
Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $ \gamma_t(G) $. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance 2 from it. The disjunctive total domination number, $ \gamma_t^d (G) $, is the minimum cardinality of such a set. We observe that $ \gamma_t^d (G) \ge \gamma_t (G) $. A leaf of $G$ is a vertex of degree 1, while a support vertex of $G$ is a vertex adjacent to a leaf. We show that if $T$ is a tree of order $n$ with $ \mathcal{l} $ leaves and $s$ support vertices, then $ 2(n−\mathcal{l}+3) // 5 \le \gamma_t^d (T) \le (n+s−1)//2 $ and we characterize the families of trees which attain these bounds. For every tree $T$, we show have $ \gamma_t(T) // \gamma_t^d (T) <2 $ and this bound is asymptotically tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 153-171
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Protection of Lexicographic Product Graphs
Autorzy:
Martínez, Abel Cabrera
Rodríguez-Velázquez, Juan Alberto
Powiązania:
https://bibliotekanauki.pl/articles/32304140.pdf
Data publikacji:
2022-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total weak Roman domination
secure total domination
total domination
lexicographic product
Opis:
Given a graph G with vertex set V (G), a function f : V (G) → {0, 1, 2} is said to be a total dominating function if Σu∈N(v) f(u) > 0 for every v ∈ V (G), where N(v) denotes the open neighbourhood of v. Let Vi = {x ∈ V (G) : f(x) = i}. A total dominating function f is a total weak Roman dominating function if for every vertex v ∈ V0 there exists a vertex u ∈ N(v) ∩ (V1 ∪ V2) such that the function f′, defined by f′(v) = 1, f′(u) = f(u) − 1 and f′(x) = f(x) whenever x ∈ V (G) \ {u, v}, is a total dominating function as well. If f is a total weak Roman dominating function and V2 = ∅, then we say that f is a secure total dominating function. The weight of a function f is defined to be ω(f) = Σv∈V (G) f(v). The total weak Roman domination number (secure total domination number) of a graph G is the minimum weight among all total weak Roman dominating functions (secure total dominating functions) on G. In this article, we show that these two parameters coincide for lexicographic product graphs. Furthermore, we obtain closed formulae and tight bounds for these parameters in terms of invariants of the factor graphs involved in the product.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 3; 967-984
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
An upper bound on the total outer-independent domination number of a tree
Autorzy:
Krzywkowski, M.
Powiązania:
https://bibliotekanauki.pl/articles/255991.pdf
Data publikacji:
2012
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
total outer-independent domination
total domination
tree
Opis:
A total outer-independent dominating set of a graph G = (V (G),E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G, denoted by [formula], is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l leaves and s support vertices we have [formula], and we characterize the trees attaining this upper bound.
Źródło:
Opuscula Mathematica; 2012, 32, 1; 153-158
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 26

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