Total Domination on Tree Operators

Bounds on the differentiating-total domination number of a tree

Given a graphG = (V , E)with no isolated vertex, a subset S of V is called a total dominating set of G if every vertex in V is adjacent to a vertex in S. A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N[u] ∩ S ≠ N[v] ∩ S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-...

Graffiti.pc on the total domination number of a tree

The total domination number of a simple, undirected graph G is the minimum cardinality of a subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In 2007 Graffiti.pc, a program that makes graph theoretical conjectures, was used to generate conjectures on the total domination number of connected graphs. More recently, the program was used to generate conjectur...

On total domination and support vertices of a tree

The total domination number γt(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of C...

A note on the total domination number of a tree

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number t(G). We show that for a nontrivial tree T of order n and with ` leaves, t(T ) > (n+2 `)=2, and we characterize the trees attaining this lower bound. Keywords: total domination, trees. AMS subject classi...

Total domination and total domination subdivision numbers