Perfectly relating the domination, total domination, and paired domination numbers of a graph

A characterization relating domination, semitotal domination and total Roman domination in trees

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γt. The maximal size of an inclusionwise minimal total dominating set, the upper total domination num...

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...

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...