Thanks for your responses.
I have just found this in PEU by Micheal Swan,page 278 and 279.
2- -ing form with its own object
The -ing form subject, object or complement is still a verb, and can have its own object.
- Smoking cigarettes is bad for you.
- I hate packing suitcases.
- My favourite activity is reading poetry.
3- determiners and possessives with -ing forms
We can often use determiners (for example the, my, this) with -ing forms.
- the opening of Parliament.
- Does my smoking annoy you?
- I don't mind your going without me.
- I hate all this useless arguing.
Possessives 's forms are also possible.
- John's going to sleep during the wedding was rather enbarrassing.
- She was angry at Lina's trying to lie to her.
Note that possessives and pronouns are not used before -ing forms if it is already clear who is being talked about.
- Thank you for waiting.(not Thank you for your waiting.)
When an -ing form is used with an article, it cannot usually have a direct object. Instead, we can use an of-structure.
- the smoking of cigarettes.(not the smoking cigarettes.)
4- object pronouns before -ing forms
In an informal style it is more common to use object forms (like me, John) instead of possessives(my, John's) with -ing forms, especially when these come after a verb or a preposition.
- I don't mind you going without me.
- She was angry at Lina trying to lie to her.
From the above, I think A and C are correct in this sentence.
- We objected to _____ up our garden.
A. their cat's digging
B. their cat to dig
C. their cat digging
D.their cat's which dig