We present optimized source galaxy selection schemes for measuring cluster weak lensing (WL) mass profiles unaffected by cluster member dilution from the Subaru Hyper Suprime-Cam Strategic Survey Program (HSC-SSP). The ongoing HSC-SSP survey will uncover thousands of galaxy clusters to $zlesssim1.5$. In deriving cluster masses via WL, a critical source of systematics is contamination and dilution of the lensing signal by cluster {members, and by foreground galaxies whose photometric redshifts are biased}. Using the first-year CAMIRA catalog of $sim$900 clusters with richness larger than 20 found in $sim$140 deg$^2$ of HSC-SSP data, we devise and compare several source selection methods, including selection in color-color space (CC-cut), and selection of robust photometric redshifts by applying constraints on their cumulative probability distribution function (PDF; P-cut). We examine the dependence of the contamination on the chosen limits adopted for each method. Using the proper limits, these methods give mass profiles with minimal dilution in agreement with one another. We find that not adopting either the CC-cut or P-cut methods results in an underestimation of the total cluster mass ($13pm4%$) and the concentration of the profile ($24pm11%$). The level of cluster contamination can reach as high as $sim10%$ at $Rapprox 0.24$ Mpc/$h$ for low-z clusters without cuts, while employing either the P-cut or CC-cut results in cluster contamination consistent with zero to within the 0.5% uncertainties. Our robust methods yield a $sim60sigma$ detection of the stacked CAMIRA surface mass density profile, with a mean mass of $M_mathrm{200c} = (1.67pm0.05({rm {stat}}))times 10^{14},M_odot/h$.
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