We consider the problem of maintaining the strongly connected components (SCCs) of an n-nodes and medges directed graph that undergoes a sequence of edge deletions. Recently, in SODA 2011, Łącki presented a deterministic algorithm that preprocess the graph in O(mn) time and creates a data structure that maintains the SCCs of a graph under edge deletions with a total update time of O(mn). The da...

Levialdl [6J introduced a parallel synchronous algorithm for counting the number of connected components in a binary image embedded in an n X n mesh of processors that runs in time O(n). We describe a parallel asynchronous algorithm for the same problem achieving the same time bound. ·Research done while visiting the International Computer Science Institute, Berkeley, California, research suppo...

The Common Connected Problem (CCP) consists in identifying common connected components in two or more graphs on the same vertices (or reduced to). More formally, let G1(V,E1) and G2(V,E2) be two such graphs and let V ′ ⊂ V . If G1[V ] and G2[V ] are both connected, V ′ is said a common connected component. The CCP problem is the identification of maximal (for the inclusion order) such component...

We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.4). This result extends theorem of Shelah from [14]. We consider also in this context the multiplicative and the additive groups of an infinite field.