  The existence of basket, flat plumbing and flat plumbing basket surfaces of a
link was first proven from a braid representative of the link. In the present
article, we show the existence of such surfaces from an induced graph of the
link. Consequently, we define the basket number, flat plumbing number and flat
plumbing basket number of a link. Then we provide several upper bounds for
these plumbing numbers and study the relation between these plumbing numbers
and the genera of links.
