In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...

A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V −S is adjacent to a vertex in S. A dominating set which intersects every maximum independent set in G is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of G and is denoted by γ...

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. 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.

We present a unified approach to Doob’s Lp maximal inequalities for 1 ≤ p < ∞. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob’s maximal inequalities. These are best possib...

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...