16.2: Basics
a path that consists of a single vertex
a closed path
a nontrivial cycle that is also a trail
If \(G\) contains vertices \(v,v'\) and edge \(e = \{v,v'\}\text{,}\) then \(v,e,v',e,v\) is a nontrivial cycle which is not proper.
contains no proper cycles
synonym for acyclic graph
a connected, acyclic graph
The graph in Figure \(\PageIndex{1}\) is acyclic. Each of its connected components is a tree.
In Worked Example 15.2.3 , we attempted to determine all possible trails from one node to another in a given graph. The graph in Figure 15.2.2 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.
- 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.