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This paper proposes the concept of safe carrier-sensing range under the cumulative interference model that guarantees interference-safe (also known as hidden-node-free) transmissions in CSMA networks. Compared with a previous related concept of safe carrier-sensing range under the commonly assumed but less realistic pairwise interference model, we show that the safe carrier-sensing range under the cumulative interference model is larger by a constant multiplicative factor. For example, the factor is 1.4 if the SINR requirement is 10 dB and the path-loss exponent is 4 in a noiseless case. We further show that the concept of a safe carrier-sensing range, although amenable to elegant analytical results, is inherently not compatible with the conventional power-threshold carrier-sensing mechanism (e.g., that used in IEEE 802.11). Specifically, the absolute power sensed by a node in the conventional carrier-sensing mechanism does not contain enough information for the node to derive its distances from other concurrent transmitting nodes. We show that, fortunately, a new carrier-sensing mechanism called Incremental-Power Carrier-Sensing (IPCS) can realize the carrier-sensing range concept in a simple way. Instead of monitoring the absolute detected power, the IPCS mechanism monitors every increment in the detected power. This means that IPCS can separate the detected power of every concurrent transmitter, and map the power profile to the required distance information. Our extensive simulation results indicate that IPCS can boost spatial reuse and network throughput by up to 60 percent relative to the conventional carrier-sensing mechanism under the same carrier-sensing power thresholds. If we compare the maximum throughput in the interference-free regime, the throughput improvement of IPCS is still more than 15 percent. Last but not least, IPCS not only allows us to implement the safe carrier-sensing range, but also ties up a loose end in many other prior theoretical work- that implicitly used a carrier-sensing range (interference-safe or otherwise) without an explicit design to realize it.