A description of paths on three vertices (3-paths) in normal plane maps (NPMs) is tight if no parameter of this description can be improved and no term dropped.We know that there are precisely seven tight descriptions of 3-paths in triangle-free NPMs: {(5,3,6),(4,3,7)}, {(3,5,3),(3,4,4)}, {(5,3,6),(3,4,3)}, {(3,5,3),(4,3,4)}, (5,3,7), (3,5,4), and (5,4,6).The problem of describing all tight descriptions of 3-paths in arbitrary NPMs is widely open. It is only known that there are precisely three tight descriptions consisting of one triplet of restrictions on the vertex degrees of 3-paths: (5,∞,6), (5,10,∞), or (10,5,∞).If NPMs are allowed to have K4−e as subgraphs, then it can happen that every 3-path goes through a vertex of unbounded degree: for example, we delete every other horizontal edge from a double 2n-pyramid. This fact explains the appearance of ∞ in the above description.The purpose of this paper is to prove that there are precisely four tight one-term descriptions of 3-paths in normal plane maps without K4−e: (10,5,20), (10,6,15), (5,10,15), and (5,15,6). In particular, the same description holds for 3-polytopes.