We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most c, where c ≈ 1.6702. The best known lower bound for this number, due to Tomescu, is ad, where d = 10 1 5 ≈ 1.5849 and a is a constant.

Dynkin’s classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to matrix Lie superalgebras, i.e., the Lie subsuperalgebras of gl(p|q).

The trace of a set F on a another set X is F |X = F ∩ X and the trace of a family F of sets on X is FX = {F |X : F ∈ F}. In this note we prove that if a k-uniform family F ⊂ ([n] k ) has the property that for any k-subset X the trace F|X does not contain a maximal chain (a family C0 ⊂ C1 ⊂ ... ⊂ Ck with |Ci| = i), then |F| ≤ ( n−1 k−1 ) . This bound is sharp as shown by {F ∈ ([n] k ) , 1 ∈ F}. ...

A dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \D has at least one neighbour in D. Moreover if D is an independent set, i.e. no vertices in D are pairwise adjacent, then D is said to be an independent dominating set. Finding a minimum independent dominating set in a graph is an NP-hard problem. We give an algorithm computing a minimum independent dom...

Let n > 1 be an integer and let G be a graph of order p. A set 9 of vertices of G is a total n-dominating set of G if every vertex of V(G) is within distance n from some vertex of .P’ other than itself. The minimum cardinal@ among all total n-dominating sets of G is called the total n-domination number and is denoted by “J:(G). A set S of vertices of G is n-independent if the distance (in G) be...

For an arbitrary graph G = (V,E), let X be a graphical property that can be possessed, or satisfied by the subsets of V . For example, being a clique (maximal complete subgraph), a maximal independent set, an edge, a closed neighborhood, a minimal dominating set, etc. Let CX ={A|A ⊆ V , |A| > 1, and A possesses or satisfies property X}. A set S is an X cover (or X free) if A ∩ S (or A ∩ (V − S)...

We characterize the complexity of a number of basic optimization problems for unit disk graphs speciied hierarchically as in BOW83, LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered. These problems include minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum ...

We study the existence and the number of k-dominating independent sets in certain graph families. While the case k = 1 namely the case of maximal independent sets which is originated from Erdős and Moser is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of kdominating independent sets in n-vertex graphs is bet...