grammar

Please help me.
In the sentence " The chasing of dogs is fun."
I think chasing is gerund.
But, I would like to know it is correct or not if i assume the grammar structure of part of sentence "the chasing of dogs" as "noun preposition noun phrase".

Similarly, in the sentence " I caused a problem by inviting John."
What is the grammar structure of the part of sentence " inviting John"?

Sorry ! my question may confuse you. Let me explain in detail. In my work, I have to classify which part of a sentence is matched with among the following grammar patterns.
subject aux noun
subject passive-verb
subject active-verb
subject verb infinitive

noun aux dobj
gerund dobj
infinitive dobj
verb infinitive dobj
active-verb dobj

noun preposition noun phrase
active-verb prep noun phrase
passive-verb prep noun phrase
infinitive prep noun phrase

I have problems in deciding which part matches with which grammar structure for some sentence.

In the sentence "Trying to hit Susan was stupid "
What is grammar structure of "trying to hit" ?

What is the difference between noun and noun phrase.
Can we assume a single word such as "Tuesday" as noun phrase?