We show that theMinimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length boun...

Let R be an o-minimal expansion of a group in language which Th(R) eliminates quantifiers, and let C predicate for valuational cut R. We identify condition that implies quantifier elimination Th(R,C) the expanded by small number constants, which, turn, is implied having being universally axiomatizable. The applies example case when convex subring field its residue o-minimal.

The Reliability analysis of information and automation systems has to cope with complex system structures and a large number of different components. Adapted to these requirements, the Markovian minimal cut approach has been developed. The Markovian minimal cut approach combines the advantages of two well-known approaches, the minimal cut and the Markovian path approach. The minimal cut approac...

Let S be a cut of a simple connected graph G. If S has no proper subset that is a cut, we say S is a minimal cut of G. To a minimal cut S, a connected component of G − S is called a fragment. And a fragment with no proper subset that is a fragment is called an end. In the paper ends are characterized and it is proved that to a connected graph G = (V, E), the number of its ends Σ ≤ |V (G)|.

For a connected graph G = (V,E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. A natural condition is to impose that for any u ∈ U the subgraph induced by (V \U)∪{u} is connected. In that case U is called a minimal disconnected cut. We show that the problem of testing whether a graph has a minimal disconnected cut i...