Displaying similar documents to “Restrained domination in unicyclic graphs”

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let G = ( V , E ) be a simple graph. A 3 -valued function f V ( G ) { - 1 , 0 , 1 } is said to be a minus dominating function if for every vertex v V , f ( N [ v ] ) = u N [ v ] f ( u ) 1 , where N [ v ] is the closed neighborhood of v . The weight of a minus dominating function f on G is f ( V ) = v V f ( v ) . The minus domination number of a graph G , denoted by γ - ( G ) , equals the minimum weight of a minus dominating function on G . In this paper, the following two results are obtained. (1) If G is a bipartite graph of order n , then γ - ( G ) 4 n + 1 - 1 - n . (2) For any negative integer k and any positive integer...

Signed total domination number of a graph

Bohdan Zelinka (2001)

Czechoslovak Mathematical Journal

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The signed total domination number of a graph is a certain variant of the domination number. If v is a vertex of a graph G , then N ( v ) is its oper neighbourhood, i.e. the set of all vertices adjacent to v in G . A mapping f : V ( G ) { - 1 , 1 } , where V ( G ) is the vertex set of G , is called a signed total dominating function (STDF) on G , if x N ( v ) f ( x ) 1 for each v V ( G ) . The minimum of values x V ( G ) f ( x ) , taken over all STDF’s of G , is called the signed total domination number of G and denoted by γ s t ( G ) . A theorem stating lower bounds for γ s t ( G ) is...

Distance in stratified graphs

Gary Chartrand, Lisa Hansen, Reza Rashidi, Naveed Sherwani (2000)

Czechoslovak Mathematical Journal

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A graph G is stratified if its vertex set is partitioned into classes, called strata. If there are k strata, then G is k -stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color X in a stratified graph G , the X -eccentricity e X ( v ) of a vertex v of G is the distance between v and an X -colored vertex furthest from v . The minimum X -eccentricity among the vertices of G is the X -radius r a d X G of G ...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class C f of all finite undirected graphs. If G is a graph from C f , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Domination in partitioned graphs

Zsolt Tuza, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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Let V₁, V₂ be a partition of the vertex set in a graph G, and let γ i denote the least number of vertices needed in G to dominate V i . We prove that γ₁+γ₂ ≤ [4/5]|V(G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ₁+γ₂ for graphs with minimum valency δ, and conjecture that γ₁+γ₂ ≤ [4/(δ+3)]|V(G)| for δ ≤ 5. As δ gets large, however, the largest...

A note on the independent domination number of subset graph

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

Czechoslovak Mathematical Journal

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The independent domination number i ( G ) (independent number β ( G ) ) is the minimum (maximum) cardinality among all maximal independent sets of G . Haviland (1995) conjectured that any connected regular graph G of order n and degree δ 1 2 n satisfies i ( G ) 2 n 3 δ 1 2 δ . For 1 k l m , the subset graph S m ( k , l ) is the bipartite graph whose vertices are the k - and l -subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i ( S m ( k , l ) ) and...