Displaying similar documents to “Partitioning a graph into a dominating set, a total dominating set, and something else”

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...

On the Totalk-Domination in Graphs

Sergio Bermudo, Juan C. Hernández-Gómez, José M. Sigarreta (2018)

Discussiones Mathematicae Graph Theory

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Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter,...

Characterization Of Super-Radial Graphs

K.M. Kathiresan, G. Marimuthu, C. Parameswaran (2014)

Discussiones Mathematicae Graph Theory

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In a graph G, the distance d(u, v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius, r(G), of the graph and the maximum eccentricity is called the diameter, d(G), of the graph. The super-radial graph R*(G) based on G has the vertex set as in G and two vertices u and v are adjacent in R*(G) if the distance between them in G is...

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.

Effect of edge-subdivision on vertex-domination in a graph

Amitava Bhattacharya, Gurusamy Rengasamy Vijayakumar (2002)

Discussiones Mathematicae Graph Theory

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Let G be a graph with Δ(G) > 1. It can be shown that the domination number of the graph obtained from G by subdividing every edge exactly once is more than that of G. So, let ξ(G) be the least number of edges such that subdividing each of these edges exactly once results in a graph whose domination number is more than that of G. The parameter ξ(G) is called the subdivision number of G. This notion has been introduced by S. Arumugam and S. Velammal. They have conjectured that for any...