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The randomness enhancement of physical chaos generated by slave vertical-cavity surface-emitting lasers (S-VCSEL) subject to dual-path polarization-preserved optical injection (DP-PPOI) from single master VCSEL (M-VCSEL) with variable-polarization optical feedback (VPOF) is investigated numerically. The randomness of chaotic signals is evaluated quantitatively by an information-theory-based quantifier, the permutation entropy. The randomness properties for S-VCSEL with DP-PPOI and S-VCSEL with single-path PPOI are compared, as well as the effects of injection strength, frequency detuning, and VPOF are considered. It is shown that, the PE values for S-VCSELs with two different injection schemes are both much higher than those for M-VCSEL, and increase initially and then decrease until they saturate at a constant level. The region of injection parameter space contributing to randomness-enhanced chaos in S-VCSEL can be greatly broadened by adopting DP-PPOI. The generation of randomness-enhanced chaos via photonic approach is highly desirable for high-speed random number generators based on chaotic VCSELs.