Answer:
Your answer would be that is tugging is the main verb in the sentence and fastened is the participle.
Step-by-step explanation:
In is tugging, to combination of the auxiliary verb to be + -ing forms the present progressive tense. This tense indicates continous action; it is used to refer to an event going on at the moment of speaking. The verb to be helps the present participle of the main verb to form the present progressive tense.
On the contrary, fastened to the pole functions as a past participle phrase modifying the noun chain. The past participle fastened takes the PP object to the pole. What is more, participle phrases always function as adjectives, adding description to the sentence. This participle phrase helps identify which chain the speaker is referring to; it is essential. If it was taken out, the meaning of the sentence would be altered (see II).
I) The dog is tugging at the chain fastened to the pole
II) The dog is tugging at the chain
Although the meaning would be altered, the sentence would still be grammatical. This would not be the case if the participle functioning as the main verb was removed, making clear the difference between a partiple behaving as main verb (III) and a participle functioning as a modifier (II).
III) *The dog at the chain fastened to the pole