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Strictly singular operators on variable exponent (or Nakano) function spaces $L^{p\left(·\right)}$ are characterized in terms of being ${\ell }_q$-strictly singular for the values $q$ in the essential range $R_{p\left(·\right)}$ of the exponent function. This extends a result of L. Weiss [On perturbations of Fredholm operators in $L_p$-spaces, Proc. Amer. Math. Soc. 67 (1977), 287-292] for $L^p$-spaces.