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

Inequalities Involving Independence Domination , / - Domination , Connected and Total / - Domination Numbers

Let / be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an /-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 /-dominating set is denned to be the /-domination number, denoted by 7/(G). In a similar way one can define the connected and total /-domination numbers 7 C| /(G) and...

On graphs with equal total domination and connected domination numbers

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...

Total domination and total domination subdivision numbers

Complexity of Comparing the Domination Number to the Independent Domination, Connected Domination, and Paired Domination Numbers

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V...

A note on domination and independence-domination numbers of graphs∗

Vizing’s conjecture is true for graphs G satisfying γ(G) = γ(G), where γ(G) is the domination number of a graph G and γ(G) is the independence-domination number of G, that is, the maximum, over all independent sets I in G, of the minimum number of vertices needed to dominate I . The equality γ(G) = γ(G) is known to hold for all chordal graphs and for chordless cycles of length 0 (mod 3). We pro...