On weighted efficient total domination

On weighted efficient total domination

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational comp...

Weighted efficient domination problem on some perfect graphs

Given a simple graph G = (V; E), a vertex v ∈ V is said to dominate itself and all vertices adjacent to it. A subset D of V is called an e cient dominating set of G if every vertex in V is dominated by exactly one vertex in D. The e cient domination problem is to 3nd an e cient dominating set of G with minimum cardinality. Suppose that each vertex v ∈ V is associated with a weight. Then, the we...

Total domination and total domination subdivision numbers

Hypo-efficient domination and hypo-unique domination

For a graph $G$ let $gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets...