Neighborhood of a Vertex | Open and Closed Neighborhoods, Graph Theory

What is the neighborhood of a vertex? Remember that the neighbors of a vertex are its adjacent vertices. So what do you think its neighborhood is? We’ll be going over neighborhoods, both open neighborhoods and closed neighborhoods, and an alternative definition of neighborhood, in today’s video graph theory lesson!

The neighborhood of a vertex v is the set containing all vertices adjacent to v. This is also called an open neighborhood. Unless otherwise stated, a neighborhood is assumed to be open. The closed neighborhood of a vertex v is the set containing all vertices adjacent to v as well as v itself. So if v is adjacent to a, b, and c, then the open neighborhood of v, written like this: N(v), is {a, b, c} and the closed neighborhood of v, written as N[v], is {a, b, c, v}.

It follows from the definition that the cardinality of the open neighborhood of a vertex is equal to the vertex’s degree.

Alternatively, the neighborhood of a vertex v may be defined as the subgraph induced by the set of all vertices adjacent to v. This definition extends similarly to closed neighborhoods. Watch the full lesson for more details!

SOLUTION TO PRACTICE PROBLEM:

The neighborhood of a is the set of all vertices adjacent to a, which is {c, d, f}. Remember we assume that the open neighborhood is desired since the question didn’t specify otherwise. The closed neighborhood of a, if desired, is {c, d, f, a}.

If you're taking a course in Graph Theory, or preparing to, you may be interested in the textbook that introduced me to Graph Theory: “A First Course in Graph Theory“ by Gary Chartrand and Ping Zhang. It’s a wonderful text! You can purchase this book through my Amazon affiliate link below! Using the affiliate link costs you nothing extra, and helps me continue to work on Wrath of Math!

PURCHASE "A First Course in Graph Theory": https://amzn.to/31hgvvJ



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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