Data Structures and Algorithms
with Object-Oriented Design Patterns in C++ 
The breadth-first traversal enqueues each node in the graph at most once. When a node is dequeued, all the edges emanating from that node are considered. Therefore, a complete traversal enumerates every edge in the graph.
The actual running time of the breadth-first traversal routine depends on the graph representation scheme used. The worst-case running time for the traversal of a graph represented using an adjacency matrix is
When adjacency lists are used, the worst case running time for the breadth-first traversal routine is
If the graph is sparse, then 
.
Therefore, if a sparse graph is represented using adjacency lists
and if 
,
the worst-case running time of the breadth-first traversal
is just 
.
Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.