Total Dominating Set

For a graph G and a subset S^t of the vertex set V(G), denote by N_G^t[S^t] the set of vertices in G which are adjacent to a vertex in S^t. If N_G^t[S^t]=V(G), then S^t is said to be a total dominating set (of vertices in G). 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 S^t, the members of S^t are required to themselves be adjacent to a vertex in S^t, whereas is an ordinary dominating set S, the members of S may be either in S itself or adjacent to vertices in S.


For example, in the Petersen graph illustrated above, the set S={1,2,9} is a (minimum) dominating set (left figure), while S^t={4,8,9,10} is a (minimum) total dominating set (right figure).

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

See also

Dominating Set, Total Domination Number

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

Cite this as:

Weisstein, Eric W. "Total Dominating Set." From MathWorld--A Wolfram Web Resource.

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