Until recently, debugging still takes almost 70% of the time in software engineering. The conventional debugging process, based on setting breakpoints and inspecting the states on them, remains the most common and useful way to detect faults. The efficiency of debugging differs a lot as both the selection and inspection of breakpoints are up to programmers. This paper presents a novel breakpoint generating approach based on a new concept named minimum debugging frontier sets (abbr. MDFS). A debugging frontier set describes a set of trace points, which have the ability of bug isolation, and a MDFS is the one with minimum size. Benefiting from the ability of bug isolation, the error suspicious domain will always be narrowed down to one side of the MDFS no matter the state of MDFS is proven correct or not. Breakpoints generated on the basis of MDFS also make the statement instances to be inspected at each breakpoint at the minimum. The paper also establishes a set of criterions to judge the quality of breakpoints. Empirical result indicates that breakpoints generated through this approach not only require low inspecting cost, but also have the ability to accelerate the efficiency of debugging. It also shows that this MDFS-based debugging prototype performs better than the state-of-art fault-localization techniques on the Siemens Suite.