Total Dominating Set

For a graph  and a subset  of the vertex set , denote by  the set of vertices in  which are adjacent to a vertex in . If , then  is said to be a total dominating set (of vertices in ). Because members of a total dominating set must be adjacent to another vertex, total dominating sets are not defined for graphs having an isolated vertex.

The total dominating set differs from the ordinary dominating set in that in a total dominating set , the members of  are required to themselves be adjacent to a vertex in , whereas is an ordinary dominating set , the members of  may be either in  itself or adjacent to vertices in .

For example, in the Petersen graph illustrated above, the set  is a (minimum) dominating set (left figure), while  is a (minimum) total dominating set (right figure).

The size of a minimum total dominating set  is called the total domination number.

REFERENCES

Henning, M. A. and Yeo, A. Total Domination in Graphs. New York: Springer, 2013.