Let G be a general graph with exactly two vertices x and y of odd degree. Let G^* be the general

Let G be a general graph with exactly two vertices x and y...

Key Concepts

Graph Connectivity

Graph connectivity is a foundational concept in graph theory that determines whether there is a path between every pair of vertices in a graph. A graph is said to be connected if for any two vertices, there exists a sequence of edges (a path) that joins them. This concept is crucial for understanding how the structure of a graph influences its behavior under various operations, such as adding or removing edges.

Edge Addition and its Effects

The process of adding an edge to a graph is a common operation that can alter the graph's structure. Generally, adding an edge cannot disrupt the existing connectivity of a graph; rather, it can only maintain or enhance it by providing additional paths between vertices. This idea is essential when analyzing how modifications, like adding an edge between two vertices, impact overall graph connectivity.

Degree Parity

Degree parity refers to whether the number of edges incident on a vertex (its degree) is odd or even. This concept is significant in various theorems, such as those concerning Eulerian trails, where a connected graph having exactly two vertices of odd degree indicates the existence of an Eulerian path. Understanding degree parity is important in analyzing connectivity, especially when the structure of a graph is altered by operations that might affect vertex degrees.

Bridges (Cut Edges)

A bridge or cut edge is an edge whose removal increases the number of connected components in a graph. This concept is important because it identifies critical connections in the graph; if an edge is a bridge, its presence is essential for maintaining connectivity between certain parts of the graph. Evaluating whether an added edge functions as a bridge helps in determining if and how the overall connectivity of a graph changes with modifications.

Transcript

00:02 So this question, we'll try that if g is connected, then it is possible to remove vertices to disconnect g if and only if g is not complete.

00:12 So one way to approach this problem is to show that if g is complete, then we cannot remove vertices to disconnect g.

00:30 And alternatively, if g is not complete, then we can remove vertices to disconnect g, right? doing so, essentially showing that these two parts are both true will give us information needed to draw the conclusion of problem 51.

01:00 So first off, let's assume that g is a connected graph on n vertices, right? so if we remove one vertex, then we'll get a connected graph on n -minus -1 vertices, right? because if we remove one vertex, then the degree of all the remaining vertices goes down by exactly one.

01:25 So all m -minus -1 vertices will have degree m -minus 2, because the edge is connecting, connecting all pairs, all possible pairs of these n minus one remaining vertices will still be intact.

01:42 There are still be edges.

01:43 So basically, if we remove, removing j vertices then, right, gives us a complete graph on n minus j on all n minus j remaining vertices.

01:59 So this goes on forever.

02:01 And so we cannot, in conclusion, we cannot remove, cannot disconnect g by removing vertices because no matter how many we remove, we will just get a complete graph on the remaining vertices in the graph.

02:21 So the first part of the statement is complete, right? we've shown that if g is complete then we no matter what we what vertices remove, g will always remain connected...

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