A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. In this paper, we offer a survey of selected recent results on total domination in graphs. c © 2008 Elsevier B.V. All rights reserved.

Let G = (V,E) be a graph and let S & V. The set S is a dominating set of G is every vertex of V-S is adjacent to a vertex of S. A vertex v of G is called S-perfect if \N[t~]nsi = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, O(G)...

For each vertex v in a graph G, let there be associated a subgraph Hv of G. The vertex v is said to dominate Hv as well as dominate each vertex and edge of Hv. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number γFH(G). A full dom...

A subset S of vertices in a graph G is a global total dominating set, or just GTDS, if S is a total dominating set of both G and G. The global total domination number γgt(G) of G is the minimum cardinality of a GTDS of G. We present bounds for the global total domination number in graphs.

We give an 0 (nlogln) [ime algorirnm for finding a minimum independem dominating se[ in a pennmation graph. TItis improves on ilie previous D(n 3) time algorictun known for solving tllis problem [4]. .,. Dept of CompUlcr Sci., Purdue Univ., West Gf:tyelle, IN 47907. Rcso::trch ~upported by ONR. Contr:lct NOOOI-l-34-K. 0502:md NSF Gl':tnl DCR-8451393, wilh matching funds from AT&T. • o.:pt of Ma...

For a graph G = (V,E), a set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S as well as another vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper we find all graphs G satisfying γr(G) = n− 3, where n is the order of G.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set of G. Let G be a connected spanning subgraph of Ks;s, and let H be the complement of G relative to Ks;s; that is, Ks;s = G ⊕ H is a factorization of Ks;s. The graph G is k-supercritical relative...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph...

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most (n−2√n )/4+O(log n) edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.