Displaying similar documents to “On the Totalk-Domination in Graphs”

Partitioning a graph into a dominating set, a total dominating set, and something else

Michael A. Henning, Christian Löwenstein, Dieter Rautenbach (2010)

Discussiones Mathematicae Graph Theory

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A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.

On the p-domination number of cactus graphs

Mostafa Blidia, Mustapha Chellali, Lutz Volkmann (2005)

Discussiones Mathematicae Graph Theory

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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.

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.

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...

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...

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.

A Gallai-type equality for the total domination number of a graph

Sanming Zhou (2004)

Discussiones Mathematicae Graph Theory

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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.

The bondage number of graphs: good and bad vertices

Vladimir Samodivkin (2008)

Discussiones Mathematicae Graph Theory

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The domination number γ(G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with |D| = γ(G) is called a γ-set of G. A vertex x of a graph G is called: (i) γ-good if x belongs to some γ-set and (ii) γ-bad if x belongs to no γ-set. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph...

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...

Characterization of block graphs with equal 2-domination number and domination number plus one

Adriana Hansberg, Lutz Volkmann (2007)

Discussiones Mathematicae Graph Theory

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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...

Bounds on the Signed 2-Independence Number in Graphs

Lutz Volkmann (2013)

Discussiones Mathematicae Graph Theory

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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 α2s(G) of G. In this work, we mainly present upper bounds...

Lower bounds for the domination number

Ermelinda Delaviña, Ryan Pepper, Bill Waller (2010)

Discussiones Mathematicae Graph Theory

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In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.