We were experiencing a sewer smell in the house ... our handyman opened up the wall behind the bathroom and noted that the vent pipe was broken. He replaced the vent pipe with one that was smaller in diameter. Now, when we flush the toilet (Toto), flushing is slow and gurgley ... doing a #2, the toilet will flush, but from that point forward, any flushing with TP in the bowl, will not flush. We end up having to use the plunger, which takes several plunges to get the water in the toilet to drain. Is the problem occurring because of the smaller diameter vent pipe or could there be another reason?

Please advise.
Thanks,
Marshall

a vent pipe only has to be 1/2 the size of the fixture it is serving. so while reducing the vent stack may or may not violate code, it will not affect the draining of one fixture.
i'm guessing the old pipe was cast iron and the handyman smashed it to get it out and some of the broken old stack fell in ???

Some other reason. Either the toilet or the drain line is obstructed. Since it will flush once and then stop, it indicates that it is the drain line. What you had done is NOT usually "handyman" work, as indicated by his not using the proper sized pipe. He might have done something, such as drop a piece of the old pipe down the drain line, to have caused it, but if so, the remedy is going to be a lot more expensive than whatever you saved by not using a plumber in the first place.