Disconnected Threshold graph is Possible?

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I am a little confused in understanding the definition of the Threshold graph. My question is: Is a threshold graph be a disconnected graph? Or a threshold graph will always be connected.?

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A threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations:

  1. Addition of a single isolated vertex to the graph.
  2. Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.

That being said, a threshold graph is connected if the final vertex added is from the second operation; the addition of a vertex which is connected to all other vertices in a graph. When the definition mentions "isolated vertex", it means a vertex which has no edges connected to it. So if the final operation used in the construction of the graph is operation 1, then the threshold graph is disconnected.

Hope this helps!

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