  In this article bipartite planar graphs St_r are investigated, r the number
of their plane regions. Bounds for the graded Betti numbers and the projective
dimension of the quotient ring associated to such graphs are discussed. We
prove that r is related to algebraic invariants that arise from the projective
resolution of the edge ideal of the graph. We also deal with labeling methods
for certain graphs and show that graphs St_r admit a graceful numbering of
their edges.
