Intro Graph Theory: Components Connectivity

[wubie]

Hello,

I am having trouble understanding the definition of components and connectivity. Here is the definition I have been given in my text:

Two vertices of a graph that are joined by a path are said to belong to the same component of the graph. If the whole graph is one component, then it is said to be connected. Thus the components are connected subgraphs which are not contained in larger connected subgraphs.

Alright, I know that a subgraph is a subset of vertices and edges which itself forms a graph. I also know that a path is a subgraph in which no edge or vertex is repeated except for perhaps the first and last vertex.


Could someone perhaps reword the above definition of components and connectivity or provide a better definition of components and connectivity for me please? Thankyou.

I also have another question regarding paths. I know that perhaps the first and last vertex may be repeated, but does that mean that a subgraph is still a path if both the last and first vertex are repeated? Or that if a subgraph is still a path if either the first or last vertex is repeated? Or can both situations occur?

 

[wubie]

Ok. I got it. I found a webpage with definitions for graph theory.

http://math.skku.ac.kr/algebra/ct/khahn/graph/def1.htm

See connected.

Thanks to all that took the time to look at my post. Cheers.

 

[tacman]

Components and connectivity are important concepts in graph theory that help us understand the structure and relationships within a graph. A component is a connected subgraph, meaning that all the vertices in the subgraph are connected to each other through a path. This path can include any number of vertices and edges, as long as there is a connection between each vertex in the subgraph. This definition means that a component can include multiple paths, as long as they are all connected to each other.

Connectivity refers to the overall connectedness of a graph. A graph is considered connected if there is a path between any two vertices in the graph. This means that all the vertices in the graph are part of the same component. If a graph has multiple components, it is considered disconnected.

Regarding your question about paths, the first and last vertex can be repeated in a subgraph and it can still be considered a path. However, if both the first and last vertex are repeated, it is essentially just a loop and not a path. A path can also have either the first or last vertex repeated, but not both. In this case, the repeated vertex would just be a part of the path and not the start or end point. I hope this helps clarify these concepts for you!