  The $k$-core of a graph is defined as the maximal subgraph in which every
vertex is connected to at least $k$ other vertices within that subgraph. In
this work we introduce a distance-based generalization of the notion of
$k$-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in
which every vertex has at least $k$ other vertices at distance $\leq h$ within
that subgraph. We study the properties of the $(k,h)$-core showing that it
preserves many of the nice features of the classic core decomposition (e.g.,
its connection with the notion of distance-generalized chromatic number) and it
preserves its usefulness to speed-up or approximate distance-generalized
notions of dense structures, such as $h$-club.
  Computing the distance-generalized core decomposition over large networks is
intrinsically complex. However, by exploiting clever upper and lower bounds we
can partition the computation in a set of totally independent subcomputations,
opening the door to top-down exploration and to multithreading, and thus
achieving an efficient algorithm.
