Vertex Connectivity is Less than or Equal to Minimum Degree | Graph Theory Exercises

The vertex connectivity of every graph is less than or equal to its minimum degree, this is a simple upper bound on vertex connectivity. We prove this fact, and show an example, in today's graph theory video lesson.

This inequality is true because if a graph G is disconnected, then its vertex connectivity is 0, which is also the lowest its minimum degree could possibly be, so the connectivity is less than the minimum degree. If G is connected, then pick a vertex v of minimum degree, say the minimum degree is d. Delete the d neighbors of v. This will either make the graph trivial or disconnect v from the rest of the graph. Since deleting d = min degree vertices was sufficient to do this, we have by definition that the connectivity is less than or equal to the minimum degree.

Recall that the vertex connectivity of a graph is the minimum number of vertices we can delete from the graph to disconnect it or make it trivial.

Vertex Cuts and Vertex Connectivity:    • Vertex Cuts in Graphs (and a bit on C...  
Vertex Connectivity:    • Vertex Connectivity of a Graph | Conn...  
Simple Bounds on Vertex Connectivity:    • Simple Bounds on Vertex Connectivity ...  

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