Total domination subdivision numbers of trees

Total domination and total domination subdivision numbers

Domination subdivision numbers of trees

A set S of vertices of a graph G = (V, E) is a dominating set if every vertex of V (G) \ S is adjacent to some vertex in S. The domination number γ (G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ (G) is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at lea...

On the domination subdivision numbers of trees

A set D of vertices of a graph G is a dominating set if every vertex in V \D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam h...

Total domination subdivision numbers of graphs

A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

Trees with equal total domination and game total domination numbers

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...