Which of the following is a run-on sentence?
a. the house was owned by the mayor.
b. mike joined the army, and he became more disciplined during the training.
c. karen's uncle arrives tomorrow she wants to see him.
d. we like to take a walk after dinner.

C because this is two main clauses which have been put together without any punctuation. So it doesn't make any sense. To fix it you would have to put either a semicolon or a comma between tomorrow and she. 

C) Karen's uncle arrives tomorrow she wants to see him is a run-on sentence.

A) The house was owned by the mayor. / This is a perfect sentence. There is nothing wrong with it.

B) Mike joined the army, and he became more disciplined during the training. / Another perfect sentence. Comma represents the pause between. Nothing wrong with this as well.

C) Karen's uncle arrives tomorrow she wants to see him. / This sentence is a disaster. The correct way to fix this sentence would be, 'Karen's uncle arrives tomorrow. She wants to see him,' 'Karen's uncle arrives tomorrow, and she wants to see him,' or 'Karen's uncle arrives tomorrow; she wants to see him.'

D) We like to take a walk after dinner. / Nothing wrong with this sentence. It's good to hear that they are burning off those calories they just ate.