Consider the following graph. (a) What is the cost of its...

Key Concepts

Graph Theory

Graph theory is the field of mathematics and computer science that studies the properties and structures of graphs, which are mathematical representations consisting of nodes (vertices) connected by edges. It provides the foundation for understanding networks, relationships, and connections between entities, and underpins various concepts including trees, cycles, and connectivity that are critical in algorithmic graph problems.

Spanning Tree

A spanning tree of a graph is a subset of its edges that connects all the vertices together without forming any cycles, effectively covering the entire graph with the minimum number of edges needed for connectivity. It is a fundamental concept in graph theory used to simplify complex networks while preserving their connectivity.

Minimum Spanning Tree (MST)

A Minimum Spanning Tree (MST) is a spanning tree with the smallest possible sum of edge weights, ensuring the most cost-effective way to connect all vertices in a weighted graph. The concept is central to optimization problems in network design, where the goal is to minimize the total cost of connecting different nodes.

Kruskal's Algorithm

Kruskal's algorithm is a greedy algorithm used to find the minimum spanning tree of a weighted graph. It works by sorting all the edges in increasing order of weight and then adding them one by one to the growing forest, ensuring that no cycles are formed, until the spanning tree is complete. The algorithm relies on the graph's cut properties to ensure that each added edge is safe and contributes to an optimal solution.

Cut Property

The cut property is a key theoretical tool used in designing and justifying algorithms like Kruskal's. For any partition (cut) of the vertices in a graph, the lightest edge that crosses the cut is guaranteed to be part of some minimum spanning tree. This property justifies why certain edges can be safely added in a greedy manner during the construction of an MST without compromising optimality.

Transcript

00:01 We're as to find a connected weighted simple graph with the fewest edges possible that has more than one minimum spanning tree.

00:15 So we have that if the graph is simple and connected and has more than one minimum spanning tree, then of course the graph cannot, i guess i should say the weighted graph.

01:09 Simple and connected it has more than one minimum spanning tree, then the graph cannot be a tree itself.

01:18 Otherwise, we would only have one minimum spanning tree, which is the graph itself.

01:24 And since a graph on n vertices, or i guess i should say a tree on n vertices has exactly, n minus 1 edges, it follows that the weighted graph should have at least as many edges as vertices.

02:20 So for this problem, there are many different, infinitely different graphs we could pick.

02:30 That would be a solution.

02:30 For the simplest examples, suppose that the graph has three vertices.

02:40 Let's try and find a graph that has three edges.

02:46 If we can do this, then it follows that this graph has the minimum number of edges and still has more than one's name of tree.

02:57 Because if we only add two edges, then we have a tree already.

03:02 So consider if we have a graph with vertices.

03:08 A, b, and c, and because we have three edges, we obtain a cycle.

03:20 And let's label the rated graph so that each edge has the same weight.

03:31 So 1, 1, and 1.

03:38 Well, then it follows that for this graph, the graph with edges ab and b is the minimum spanning tree, as well as the graph with edges ab and ac, as well as the graph ac and bc.

04:25 So these three graphs are all minimum spanning trees...

Please give Ace some feedback