It is difficult to determine the correct consituent structure of prepositional verbs, such as rely on someone. Either on someone forms a constituent to the exclusion of rely, as in (1), or rely on form a constituent to the exclusion of someone, as in (2).
(1) Mary [VP [relies] [on John]] (to do something)
(2) Mary [VP [relies on] [John]] (to do something)
Which analysis of the constituency of prepositional verbs is better and why?
Addendum
The following observations might be helpful.
(3) I rely [on a good provider] and [on a AC3200 router] to assure my access to the internet.
(supports (1)?)(4) a. Mary relies on John to dress himself.
b. It is [John] that Mary relies on _ to dress himself.
c. *It is [on John] that Mary relies _ to dress himself.
(Bresnan 1982: 398) (supports (2)?)(5) They were relying, foolishly or at least naively, on the fifth division to cover their retreat.
(Postal 2004: 87) (supports (1)?)(6) a. John, she relied on _ to help with record-keeping at the school.
b. ?* On John, she relied _ to help with record-keeping at the school.
(supports (2)?)(7) a. Mary relied [on John] to fix the car, and Sue [on Bill] to mow the lawn.
b. *Mary relied on [John] to fix the car and Sue [Bill] to mow the lawn.
(supports (1)?)(8) a. Mary [relied on], and indeed [depended on], John.
b. ?? Mary [relied], and indeed [depended], [on John].
(supports (2)?)(9) ?*I relied on there to be a solution.
(not sure if this supports either (1) or (2)?)(10) These are people he [relied on _ ] and [lived with _ ].
(not sure if this supports either (1) or (2)?)(11) a. Mary said she would rely on John to help her...
b. * ... and rely on John she did _ to help her.
c. * ... and rely on she did _ John to help her.
d. * ... and rely she did _ on John to help her.
(not sure if this supports either (1) or (2)?)