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Nivat's conjecture is a famous open problem in symbolic dynamics. Theexistence of nonexpansive lines that when endowed with a given orientation areone-sided nonexpansive directions is at the heart of some advances. In hisPh.D. thesis, Michal Szabados conjectured that for a minimal periodicdecomposition the nonexpansive lines are exactly the lines that contain aperiod of some periodic configuration in such decomposition. In this paper, weprovide conditions where (i) Szabados's conjecture holds and (ii) a given lineis nonexpansive if and only if the same line endowed with a given orientationis a one-sided nonexpansive direction. As a corollary of our main result, weget that Nivat's conjecture holds for low convex complexity configurations ifand only if it holds for low convex complexity configurations satisfying (i)and (ii).

On Periodic Decompositions, One-sided Nonexpansive Directions and Nivat's Conjecture