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

are communication connections between pairs of computers on the Internet. The
computers and the connections between them in a single domain, like wiley.com,
form a subgraph of the Internet. If this subgraph is connected, then two users on
computers in this domain can send e-mail to one another without having their
information packets ever leave their domain. Suppose the edges of this subgraph form
a spanning tree. This implies that, if even a single connection goes down (for
example, because someone pulls a communication cable out of the back of a
computer in this domain), then this subgraph will no longer be connected.
There are a number of simple properties of trees, forests, and connected graphs.
Proposition 13.11: Let G be an undirected graph with n vertices and m edges.
• If G is connected, then m≥n − 1.
• If G is a tree, then m = n − 1.
• If G is a forest, then m≤n − 1.
We leave the justification of this proposition as an exercise (C-13.2).
13.1.1 The Graph ADT
As an abstract data type, a graph is a collection of elements that are stored at the
graph's positions—its vertices and edges. Hence, we can store elements in a graph
at either its edges or its vertices (or both). In Java, this means we can define
Vertex and Edge interfaces that each extend the Position interface. Let us
then introduce the following simplified graph ADT, which is suitable for vertex and
edge positions in undirected graphs, that is, graphs whose edges are all undirected.
Additional methods for dealing with directed edges are discussed in Section 13.4.
vertices():
Return an iterable collection of all the vertices of the graph.
edges():
Return an iterable collection of all the edges of the graph.
incidentEdges(v):
Return an iterable collection of the edges incident upon vertex v.
opposite(v,e):
Return the endvertex of edge e distinct from vertex v; an error occurs if e
is not incident on v.
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