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16.5: Spanning Trees

Last updated

Feb 19, 2022
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- 16.4: Depth-first and breadth-first searches
- 16.6: Binary Searches

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Definition: Spanning Subgraph

a subgraph that contains all the vertices of the parent graph

Definition: Spanning Tree

a spanning subgraph that is a tree

Example $\PageIndex{1}$ : Spanning trees for the complete graph $K_4$ .

Here is the complete graph with four vertices.

And here are ten different spanning trees for $K_4\text{.}$

If we carry out either of the depth-first or breadth-first search algorithms, but aren't looking for a path between specific vertices, the end result will be a spanning tree for the original graph.

Definition: Depth-first Spanning Tree

the result of performing the depth-first search algorithm on a graph, continuing until all vertices in the original graph appear in the search tree

Definition: Breadth-first Spanning Tree

the result of performing the breadth-first search algorithm on a graph, continuing until all vertices in the original graph appear in the search tree

Example $\PageIndex{2}$ : Depth-first and breadth-first spanning trees.

Figure $\PageIndex{3}$ contains depth-first and breadth-first spanning trees for the graph in Figure 16.4.2, our source of examples for depth-first search (Example 16.4.1) and breadth-first search (Example 16.4.2).

(a) Depth-first spanning tree for the graph in Figure 16.4.1.

Definition: Spanning Subgraph

Definition: Spanning Tree

Example 16.5.1\PageIndex{1}: Spanning trees for the complete graph K4K_4.

Definition: Depth-first Spanning Tree

Definition: Breadth-first Spanning Tree

Example 16.5.2\PageIndex{2}: Depth-first and breadth-first spanning trees.

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Example $\PageIndex{1}$ : Spanning trees for the complete graph $K_4$ .

Example $\PageIndex{2}$ : Depth-first and breadth-first spanning trees.