The (classical) domination number of a graph is the cardinality of a smallest subset of its vertex set with the property that each vertex of the graph is in the subset or adjacent to a vertex in the subset. Since its introduction to the literature during the early 1960s, this graph parameter has been researched extensively and nds application in the generic facility location problem where a sma...

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time 2(1−poly(1/c))n where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algori...

If G=(VG,EG) is a graph of order n, we call S⊆VG an isolating set if the induced by VG−NG[S] contains no edges. The minimum cardinality G called isolation number G, and it denoted ι(G). It known that ι(G)≤n3 bound sharp. A subset dominating in NG[S]=VG. domination number, γ(G). In this paper, analyze family trees T where ι(T)=γ(T), prove ι(T)=n3 implies ι(T)=γ(T). Moreover, give different equiv...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular then weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. We also prove that any unconditional logarithmic...