Trees and Spanning Definition: A tree is a connected graph G that is acyclic (contains no cycles). A spanning tree of a connected graph G is a subgraph...
Trees and Spanning
Definition:
A tree is a connected graph G that is acyclic (contains no cycles). A spanning tree of a connected graph G is a subgraph that includes all the vertices of G and no other vertices outside G.
Key Concepts:
A tree is a connected graph with n vertices, where n is the order of the graph.
A spanning tree is a subgraph of a connected graph G that contains all the vertices of G and no other vertices outside G.
A spanning tree of a graph G is unique up to a isomorphism, meaning that we can rearrange the vertices in the spanning tree in any way, as long as we preserve the edges of the graph.
A tree is a connected graph with n vertices that contains exactly n connected components.
A spanning tree can be constructed from any connected graph with n vertices.
Examples:
A tree with n vertices is a complete graph, where each vertex is connected to exactly two other vertices.
A forest is a connected graph with n vertices that is not a tree.
A spanning tree of a graph G is a subgraph of G that contains all the vertices of G and no other vertices outside G.
A spanning tree of a graph G is unique up to a isomorphism, meaning that we can rearrange the vertices in the spanning tree in any way, as long as we preserve the edges of the graph