We consider problems of the following type: given a graph G, how many edges are needed in worst case for sparse subgraph H that approximately preserves distances between set node pairs P? Examples include pairwise spanners, distance preservers, reachability etc. There has been trend area simple constructions based on hitting technique, followed by somewhat more complicated improve over bounds o...

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One ...