BreadthFirstScan

BreadthFirstScan[g, s, {event1 -> f1, event2 -> f2, ...}] performs a breadth-first scan (BFS) of the graph g starting at the vertex s and evaluates fi whenever "eventi" occurs.

BreadthFirstScan[g, {event1 -> f1, ...}] performs a breadth-first scan of the whole graph g.

Examples

BFS with vertex discovery:

BreadthFirstScan[GridGraph[{3, 3}], 1, 
  {"DiscoverVertex" -> Print}]

Please visit the official Wolfram Language Reference for more details.