In this paper, a new graph partitioning problem is introduced. The depth of each part is constrained, i.e., the node count in the longest path of the corresponding sub-graph is no more than a predetermined positive integer value p. An additional constraint is enforced such that each part contains only nodes selected from consecutive levels in the graph. The problem is therefore transformed into a Depth-bounded Levelized Graph Partitioning (DLGP) problem, which is solved optimally using a dynamic programming algorithm. As an example application, we have shown that DLGP can effectively generate timing-correct circuit solutions for Single Flux Quantum (SFQ) logic, which is a magnetic-pulse-based, gate-level pipelined superconductive computing fabric. Experimental results confirm that DLGP generates circuits with considerably lower path balancing overheads compared with a baseline full-path-balancing approach. For example, the balancing overhead (a critical measure of quality metric) for the SFQ circuit realization in terms of D-Flip-Flop count is reduced by 3.61 times on average for 10 benchmark circuit, given p=5.