In this letter, we present a new scheme to find small fundamental instantons (SFIs) of regular low-density parity-check (LDPC) codes for the linear programming (LP) decoding over the binary symmetric channel (BSC). Based on the fact that each instanton-induced graph (IIG) contains at least one short cycle, we determine potential instantons by constructing possible IIGs which contain short cycles and additional paths connected to the cycles. Then we identify actual instantons from potential ones under the LP decoding. Simulation results on some typical LDPC codes show that our scheme is effective, and more instantons can be obtained by the proposed scheme when compared with the existing instanton search method.