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Section 5.4 Paths
Subsection 5.4.1 Paths
Definition 5.4.1 . Walk.
A graph is a
walk if and only if the vertices can be labeled
\(v_0, v_1, \ldots, v_k\) such that
\(v_i, v_{i+1}\) is an edge.
Definition 5.4.2 . Trail.
A graph is a
trail if and only if it is a walk such that no edge is used twice.
Definition 5.4.3 . Path.
A graph is a
path if and only if it is a walk such that no vertex is used twice.
A path on
\(n\) vertices is denoted
\(P_n.\) Please note that these terms (
walk ,
trail , and
path ) vary widely including reversing the names, so always check the definition in the article, book, or other material you are reading.
Definition 5.4.4 . Eulerian Trail.
A trail is
Eulerian if and only if it uses every edge in the graph.
Definition 5.4.5 . Hamiltonian Path.
A path is
Hamiltonian if and only if it uses every vertex in that graph.
Subsection 5.4.2 Practice
Checkpoint 5.4.6 .
Draw a path of length 5. Note this is denoted
\(P_5.\)
Figure 5.4.7. Graph Containing Walks, Trails, and Paths
Checkpoint 5.4.8 .
Checkpoint 5.4.9 .
Checkpoint 5.4.10 .
Checkpoint 5.4.11 .
Subsection 5.4.3 Connected
Definition 5.4.12 . Vertex Connected.
A graph
\(G\) is connected if and only if for every pair of vertices
\(v,w\) there exists a path from
\(v\) to
\(w.\)
Definition 5.4.13 . Vertex \(n\) -connected.
A graph
\(G\) is
\(n\) -connected if and only if removing any
\(n-1\) vertices does not disconnect the graph.
Subsection 5.4.4 Practice
Checkpoint 5.4.14 .
Checkpoint 5.4.15 .
Explain why every complete graph is connected.
Checkpoint 5.4.16 .
Explain why every complete bipartite graph is connected.
Checkpoint 5.4.17 .
Determine if every bipartite graph must be connected.
Checkpoint 5.4.18 .
Checkpoint 5.4.19 .
If a graph is
\(n\) -connected what does this say about the minimum of the number of paths between any two vertices?