Displaying similar documents to “Domination in Analysis”

Efficient (j,k)-domination

Robert R. Rubalcaba, Peter J. Slater (2007)

Discussiones Mathematicae Graph Theory

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A dominating set S of a graph G is called efficient if |N[v]∩ S| = 1 for every vertex v ∈ V(G). That is, a dominating set S is efficient if and only if every vertex is dominated exactly once. In this paper, we investigate efficient multiple domination. There are several types of multiple domination defined in the literature: k-tuple domination, {k}-domination, and k-domination. We investigate efficient versions of the first two as well as a new type of multiple domination.

Trees with equal restrained domination and total restrained domination numbers

Joanna Raczek (2007)

Discussiones Mathematicae Graph Theory

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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...

Domination Parameters of a Graph and its Complement

Wyatt J. Desormeaux, Teresa W. Haynes, Michael A. Henning (2018)

Discussiones Mathematicae Graph Theory

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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.

Hereditary domination and independence parameters

Wayne Goddard, Teresa Haynes, Debra Knisley (2004)

Discussiones Mathematicae Graph Theory

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For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for arbitrary properties P that are closed under disjoint unions and subgraphs.

Note on the split domination number of the Cartesian product of paths

Maciej Zwierzchowski (2005)

Discussiones Mathematicae Graph Theory

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In this note the split domination number of the Cartesian product of two paths is considered. Our results are related to [2] where the domination number of Pₘ ☐ Pₙ was studied. The split domination number of P₂ ☐ Pₙ is calculated, and we give good estimates for the split domination number of Pₘ ☐ Pₙ expressed in terms of its domination number.

Trees with equal total domination and total restrained domination numbers

Xue-Gang Chen, Wai Chee Shiu, Hong-Yu Chen (2008)

Discussiones Mathematicae Graph Theory

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For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both ⟨S⟩ has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and ⟨V(G)-S⟩ has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination...

Various Bounds for Liar’s Domination Number

Abdollah Alimadadi, Doost Ali Mojdeh, Nader Jafari Rad (2016)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a dominating set if Uv∈S N[v] = V , where N[v] is the closed neighborhood of v. Let L ⊆ V be a dominating set, and let v be a designated vertex in V (an intruder vertex). Each vertex in L ∩ N[v] can report that v is the location of the intruder, but (at most) one x ∈ L ∩ N[v] can report any w ∈ N[x] as the intruder location or x can indicate that there is no intruder in N[x]. A dominating set L is called a liar’s dominating set if every v ∈ V...

A Note on Non-Dominating Set Partitions in Graphs

Wyatt J. Desormeaux, Teresa W. Haynes, Michael A. Henning (2016)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating)...

On the Totalk-Domination in Graphs

Sergio Bermudo, Juan C. Hernández-Gómez, José M. Sigarreta (2018)

Discussiones Mathematicae Graph Theory

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Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter,...

The Signed Total Roman k-Domatic Number Of A Graph

Lutz Volkmann (2017)

Discussiones Mathematicae Graph Theory

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Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → {−1, 1, 2} such that Ʃu2N(v) f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed total Roman k-dominating functions on G with the property that Ʃdi=1 fi(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating...

Total domination subdivision numbers of graphs

Teresa W. Haynes, Michael A. Henning, Lora S. Hopkins (2004)

Discussiones Mathematicae Graph Theory

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A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. First we establish bounds on the total domination subdivision number...

Remarks on restrained domination and total restrained domination in graphs

Bohdan Zelinka (2005)

Czechoslovak Mathematical Journal

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The restrained domination number γ r ( G ) and the total restrained domination number γ t r ( G ) of a graph G were introduced recently by various authors as certain variants of the domination number γ ( G ) of ( G ) . A well-known numerical invariant of a graph is the domatic number d ( G ) which is in a certain way related (and may be called dual) to γ ( G ) . The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions. ...

Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree

Michael A. Henning, Alister J. Marcon (2016)

Discussiones Mathematicae Graph Theory

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Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating...

Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs

Lutz Volkmann (2015)

Discussiones Mathematicae Graph Theory

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Let G be a graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions...

Graphs with equal domination and 2-distance domination numbers

Joanna Raczek (2011)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u-v) path in G. A set D ⊆ V(G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V(G) is a 2-distance dominating set if every vertex of G is at distance at most 2 from an element of D. The 2-distance domination number of G is the minimum...

Highly connected counterexamples to a conjecture on α-domination

Zsolt Tuza (2005)

Discussiones Mathematicae Graph Theory

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An infinite class of counterexamples is given to a conjecture of Dahme et al. [1] concerning the minimum size of a dominating vertex set that contains at least a prescribed proportion of the neighbors of each vertex not belonging to the set.

Total Domination Multisubdivision Number of a Graph

Diana Avella-Alaminos, Magda Dettlaff, Magdalena Lemańska, Rita Zuazua (2015)

Discussiones Mathematicae Graph Theory

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The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination...

Twin Minus Total Domination Numbers In Directed Graphs

Nasrin Dehgardi, Maryam Atapour (2017)

Discussiones Mathematicae Graph Theory

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Let D = (V,A) be a finite simple directed graph (shortly, digraph). A function f : V → {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f(N−(v)) ≥ 1 and f(N+(v)) ≥ 1 for each vertex v ∈ V. The twin minus total domination number of D is y*mt(D) = min{w(f) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for y*mt(D) in terms of the order, size and maximum and minimum in-degrees...

The Domination Number of K 3 n

John Georges, Jianwei Lin, David Mauro (2014)

Discussiones Mathematicae Graph Theory

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Let K3n denote the Cartesian product Kn□Kn□Kn, where Kn is the complete graph on n vertices. We show that the domination number of K3n is [...]