The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “On the minus domination number of graphs”

Signed total domination number of a graph

Bohdan Zelinka (2001)

Czechoslovak Mathematical Journal

Similarity:

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...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

Similarity:

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.

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

Similarity:

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by γ r ( G ) , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then γ r ( U ) ≥ ⎡ n / 3 ⎤ , and provide a characterization of graphs achieving this bound.

Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci (2001)

Discussiones Mathematicae Graph Theory

Similarity:

Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition N G ( x ) ⊆ N G [ u ] ∪ N G [ v ] , where N G [ x ] = N G ( x ) ∪ x . In the case where G is a claw-free graph, G* is equal to G². We define σ ° ₜ = m i n ∑ x ∈ S d G ( x ) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

On a problem concerning k -subdomination numbers of graphs

Bohdan Zelinka (2003)

Czechoslovak Mathematical Journal

Similarity:

One of numerical invariants concerning domination in graphs is the k -subdomination number γ k S - 11 ( G ) of a graph G . A conjecture concerning it was expressed by J. H. Hattingh, namely that for any connected graph G with n vertices and any k with 1 2 n < k ≦ n the inequality γ k S - 11 ( G ) ≦ 2 k - n holds. This paper presents a simple counterexample which disproves this conjecture. This counterexample is the graph of the three-dimensional cube and k = 5 .