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Section 16.2 Basics
trivial path
a path that consists of a single vertex
cycle
proper cycle
a nontrivial cycle that is also a trail
Example 16.2.1 . A nontrivial cycle that is not proper..
The cycle
\(v,e,v',e,v \) in any graph
\(G \) that contains distinct vertices
\(v,v' \) and edge
\(e = \{v,v'\} \text{.}\)
acyclic graph
contains no proper cycles
forest
synonym for
acyclic graph
tree
a connected, acyclic graph
Example 16.2.2 . A forest of trees.
The graph in
Figure 16.2.3 is acyclic. Each of its connected components is a tree.
Figure 16.2.3. A nonconnected acyclic graph.
Example 16.2.4 . Decision trees are trees.
In
Worked Example 15.2.5 , we attempted to determine all possible trails from one node to another in a given graph. The graph in
Figure 15.2.6 that we used to explore possible trails in the given graph is an example of a
decision tree — at each node we “branched out” to new possibilities in continuing the trail. As the name suggested, the connected graph we ended up with is a tree.
Proposition 16.2.5 . Subgraphs of forests.
Every subgraph of an acyclic graph is acyclic.
Every connected subgraph of an acyclic graph is a tree. In particular, each connected component of an acyclic graph is a tree.
