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In this paper, we present some theoretical and computational results concerning convexity P3 and convexity P ∗ 3 under the convex partitioning and separation perspectives. On the theoretical side, we show which aspects both convexities are equivalent to, we identify the smallest graphs that do not admit a convex partition and list an infinite class of graphs with no such partitioning, and finally, we analyze the patterns of convex separation for paths, trees, cycles, and cliques. In the computational aspect, we show the NP-completeness of the convex partitioning problem in both convexities, even restricted to particular class graphs.

ENLACES DAS CONVEXIDADES DE CAMINHOS CURTOS EM PROBLEMAS DE PARTIÇÕES E SEPARAÇÕES CONVEXAS