Displaying similar documents to “The cobondage number of a graph”

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.

Homogeneously embedding stratified graphs in stratified graphs

Gary Chartrand, Donald W. Vanderjagt, Ping Zhang (2005)

Mathematica Bohemica

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A 2-stratified graph G is a graph whose vertex set has been partitioned into two subsets, called the strata or color classes of G . Two 2 -stratified graphs G and H are isomorphic if there exists a color-preserving isomorphism φ from G to H . A 2 -stratified graph G is said to be homogeneously embedded in a 2 -stratified graph H if for every vertex x of G and every vertex y of H , where x and y are colored the same, there exists an induced 2 -stratified subgraph H ' of H containing y and a color-preserving...

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

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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.

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 .

Vertex-disjoint stars in graphs

Katsuhiro Ota (2001)

Discussiones Mathematicae Graph Theory

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In this paper, we give a sufficient condition for a graph to contain vertex-disjoint stars of a given size. It is proved that if the minimum degree of the graph is at least k+t-1 and the order is at least (t+1)k + O(t²), then the graph contains k vertex-disjoint copies of a star K 1 , t . The condition on the minimum degree is sharp, and there is an example showing that the term O(t²) for the number of uncovered vertices is necessary in a sense.