Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

• Given a start vertex s of G, computing, for every vertex v of G,a path with
the minimum number of edges between s and v, or reporting that no such path
exists.
• Computing a cycle in G, or reporting that G has no cycles.
13.4 Directed Graphs
In this section, we consider issues that are specific to directed graphs. Recall that a
directed graph (digraph), is a graph whose edges are all directed.
Methods Dealing with Directed Edges
When we allow for some or all the edges in a graph to be directed, we should add
the following two methods to the graph ADT in order to deal with edge directions.
isDirected(e): Test whether edge e is directed.
insertDirectedEdge(v, w, o): Insert and return a new directed edge
with origin v and destination w and storing element o.
Also, if an edge e is directed, the method endVertices(e) should return an array
A such that A[0] is the origin of e and A[1] is the destination of e. The running time
for the method isDirected(e) should be O(1), and the running time of the
method insertDirectedEdge(v, w, o) should match that of undirected edge
insertion.
Reachability
One of the most fundamental issues with directed graphs is the notion of
reachability, which deals with determining where we can get to in a directed graph.
A traversal in a directed graph always goes along directed paths, that is, paths
where all the edges are traversed according to their respective directions. Given
vertices u and v of a digraph
, we say that u reaches v (and v is reachable from
u) if has a directed path from u to v. We also say that a vertex v reaches an edge
(w,z) if v reaches the origin vertex w of the edge.
A digraph is strongly connected if for any two vertices u and v of , u reaches
v and v reaches u. A directed cycle of is a cycle where all the edges are
traversed according to their respective directions. (Note that may have a cycle
consisting of two edges with opposite direction between the same pair of vertices.)
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