Displaying similar documents to “Independent transversal domination in graphs”

Inequalities involving independence domination, f -domination, connected and total f -domination numbers

San Ming Zhou (2000)

Czechoslovak Mathematical Journal

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Let f be an integer-valued function defined on the vertex set V ( G ) of a graph G . A subset D of V ( G ) is an f -dominating set if each vertex x outside D is adjacent to at least f ( x ) vertices in D . The minimum number of vertices in an f -dominating set is defined to be the f -domination number, denoted by γ f ( G ) . In a similar way one can define the connected and total f -domination numbers γ c , f ( G ) and γ t , f ( G ) . If f ( x ) = 1 for all vertices x , then these are the ordinary domination number, connected domination number and total...

On domination number of 4-regular graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let G be a simple graph. A subset S ⊆ V is a dominating set of G , if for any vertex v ∈ V - S there exists a vertex u ∈ S such that u v ∈ E ( G ) . The domination number, denoted by γ ( G ) , is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n , then γ ( G ) ≤ 4 11 n .

The cobondage number of a graph

V.R. Kulli, B. Janakiram (1996)

Discussiones Mathematicae Graph Theory

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A set D of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number b c ( G ) of G to be the minimum cardinality among the sets of edges X ⊆ P₂(V) - E, where P₂(V) = X ⊆ V:|X| = 2 such that γ(G+X) < γ(G). In this paper, the exact values of bc(G) for some standard graphs are found and some bounds are obtained. Also, a Nordhaus-Gaddum...

Matchings and total domination subdivision number in graphs with few induced 4-cycles

Odile Favaron, Hossein Karami, Rana Khoeilar, Seyed Mahmoud Sheikholeslami (2010)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number s d γ ₜ ( 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 total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (Journal...

On total restrained domination in graphs

De-xiang Ma, Xue-Gang Chen, Liang Sun (2005)

Czechoslovak Mathematical Journal

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In this paper we initiate the study of total restrained domination in graphs. Let G = ( V , E ) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V - S is adjacent to a vertex in S as well as to another vertex in V - S , and every vertex in S is adjacent to another vertex in S . The total restrained domination number of G , denoted by γ r t ( G ) , is the smallest cardinality of a total restrained dominating set of G . First, some exact values and sharp bounds for γ r t ( G ) are given in Section 2....

A note on the domination number of a graph and its complement

Dănuţ Marcu (2001)

Mathematica Bohemica

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If G is a simple graph of size n without isolated vertices and G ¯ is its complement, we show that the domination numbers of G and G ¯ satisfy γ ( G ) + γ ( G ¯ ) ≤ n - δ + 2 if γ ( G ) > 3 , δ + 3 if γ ( G ¯ ) > 3 , where δ is the minimum degree of vertices in G .

A bound on the k -domination number of a graph

Lutz Volkmann (2010)

Czechoslovak Mathematical Journal

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Let G be a graph with vertex set V ( G ) , and let k ≥ 1 be an integer. A subset D ⊆ V ( G ) is called a if every vertex v ∈ V ( G ) - D has at least k neighbors in D . The k -domination number γ k ( G ) of G is the minimum cardinality of a k -dominating set in G . If G is a graph with minimum degree δ ( G ) ≥ k + 1 , then we prove that γ k + 1 ( G ) ≤ | V ( G ) | + γ k ( G ) 2 . In addition, we present a characterization of a special class of graphs attaining equality in this inequality.