Two short proofs on total domination

Two short proofs on total domination

A set of vertices of a graph G is a total dominating set if each vertex of G is adjacent to a vertex in the set. The total domination number of a graph γt (G) is the minimum size of a total dominating set. We provide a short proof of the result that γt (G) ≤ 2 3 n for connected graphs with n ≥ 3 and a short characterization of the extremal graphs.

Signed total domination on Kronecker products of two complete graphs

Given two graphs G1 and G2, the Kronecker product G1 ⊗G2 of G1 and G2 is a graph which has vertex set V (G1⊗G2) = V (G1)×V (G2) and edge set E(G1 ⊗ G2) = {(u1, v1)(u2, v2) : u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. ∗ Research was partially supported by the National Nature Science Foundation of China (No. 11171207) and the Key Programs of Wuxi City College of Vocational Technology (WXCY2012-GZ-007). † Co...

Total domination and total domination subdivision numbers

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...

On Total Domination in Graphs

LetG = (V,E) be a finite, simple, undirected graph. A set S ⊆ V is called a total dominating set if every vertex of V is adjacent to some vertex of S. Interest in total domination began when the concept was introduced by Cockayne, Dawes, and Hedetniemi [6] in 1980. In 1998, two books on the subject appeared ([11] and [12]), followed by a survey of more recent results in 2009 [15]. The total dom...