Determine a spanning tree for each of the graphs in Figures 11.15 through 11.17 .

Name: Problem 63, Chapter 11: Introductory Combinatorics
Uploaded: 2020-11-15T04:14:44-08:00
Duration: 7 min 25 s
Channel: Nick Johnson
Description: Determine a spanning tree for each of the graphs in Figures 11.15 through 11.17 .

Determine a spanning tree for each of the graphs in Figures...

Key Concepts

Connectivity

Connectivity in a graph refers to the property that there is a path between any pair of vertices. For a spanning tree to exist, the graph must be connected. This concept is crucial because only connected graphs can have spanning trees that include all vertices.

Tree

A tree is a type of graph that is connected and acyclic, meaning it contains no cycles. Trees have a hierarchical structure and are a special case of graphs. In the context of spanning trees, the tree structure is achieved by ensuring that the subgraph is free of cycles while connecting all vertices.

Cycle

A cycle in a graph is a path that starts and ends at the same vertex without repeating any edge. In spanning trees, cycles are undesirable because they violate the tree property of being acyclic. Recognizing cycles is important when constructing a spanning tree from a graph.

Graph

A graph is a collection of vertices (or nodes) connected by edges. It is a fundamental concept in discrete mathematics and computer science used to model pairwise relationships between objects. Graphs form the basis for many problems and algorithms, including those related to connectivity, traversals, and network flows.

Spanning Tree

A spanning tree of a connected graph is a subgraph that includes all the vertices of the original graph, is connected, and contains no cycles. It is used to simplify a graph while preserving its connectedness, and serves as the basis for many algorithms in network design and optimization.

Transcript

00:02 Okay, so a spanning tree of a simple graph g, where g is a subgraph of a simple graph g, where g is a simple graph of g, where g is a subgraph of g, so is a subgraph of g, that is a tree of g, that is a tree containing every vertex of g.

00:42 So a spanning tree of g is a subgraph of g that is a tree containing every vertex of g.

00:59 Containing every vertex of the graph g.

01:10 Okay, so first we're looking at k5.

01:15 So we can see, i'm not going to redraw it here, but the spanning tree, of k5.

01:24 So first looking at k5.

01:27 So the spanning tree of k5 is going to be s, which is equal to, well, here we have our node, our vertex a, which goes to our vertex b, which goes down here to our vertex c, this goes down here to our vertex c, this goes to our vertex d, and then goes over here to our vertex e.

01:55 All right.

01:57 So there is the spanning tree of k5.

02:01 Next, we're looking at k sub 4.

02:05 So you can look in the book.

02:07 This is a little kind of a crazy graph here.

02:10 But then the spanning tree of k4, okay, so the spanning tree s is going to be equal to we have our vertex a.

02:21 And then, well, so then we're going to have our vertex b, which is actually going to be here, and then our vertex c, and then our vertex d.

02:35 Okay, but then a is going to connect down to e, and then e connects up to a vertex b.

02:44 B goes down to a vertex f, and then f connects up to c, and then c connects down here to g, and then g connects up to d, and then d comes down to our vertex h.

03:06 All right.

03:07 And then, so that was part b.

03:10 And then for part c, looking at k sub 1 ,6.

03:17 Okay, and then the spanning tree here is going to be equal to s, which is equal to, well, here we have our vertex a...

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