Edit: This answer was wrong - I misremembered the result. See Alan Hatcher's answer. Of course, the references in the second paragraph still stand.

Those two operations give trivial outer automorphisms of $\pi_1(B)$, whereas the handlebody group surjects $Out(\pi_1(B))$. So Dehn twists about disks and annuli don't suffice (in fact, I think it's known that these generate the kernel of the map to $Out(\pi_1(B))$).

Suzuki gave generators for the mapping class group of a handlebody, and Wajnryb gave a presentation. See also Popescu.

Edit: This answer was wrong - I misremembered the result. See Allen Hatcher's answer. Of course, the references in the second paragraph still stand.

Those two operations give trivial outer automorphisms of $\pi_1(B)$, whereas the handlebody group surjects $Out(\pi_1(B))$. So Dehn twists about disks and annuli don't suffice (in fact, I think it's known that these generate the kernel of the map to $Out(\pi_1(B))$).

Suzuki gave generators for the mapping class group of a handlebody, and Wajnryb gave a presentation. See also Popescu.

Those two operations give trivial outer automorphisms of $\pi_1(B)$, whereas the handlebody group surjects $Out(\pi_1(B))$. So Dehn twists about disks and annuli don't suffice (in fact, I think it's known that these generate the kernel of the map to $Out(\pi_1(B))$).

Suzuki gave generators for the mapping class group of a handlebody, and Wajnryb gave a presentation. See also Popescu.

Edit: This answer was wrong - I misremembered the result. See Alan Hatcher's answer. Of course, the references in the second paragraph still stand.

Those two operations give trivial outer automorphisms of $\pi_1(B)$, whereas the handlebody group surjects $Out(\pi_1(B))$. So Dehn twists about disks and annuli don't suffice (in fact, I think it's known that these generate the kernel of the map to $Out(\pi_1(B))$).

Suzuki gave generators for the mapping class group of a handlebody, and Wajnryb gave a presentation. See also Popescu.