We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O( √ n) upper bound. Consequently, planar graphs admit 3D straight-line crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2 ) upper bound. p,q such that a ≺ b. Then, a is inserted into L ′ p,q before b. Proof: The proof easily descends from the fact that, when a vertex is inserted into a list L ′ p,q , it is appended to such a list. We now study

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... the edges of (G, f, g) relating the end-vertices of such edges to their position in L and to the floor of (G, f, g) they belong to. A visible edge is an edge of G that has one end-vertex in a list L ′ p,q , for some p and q, and one end-vertex in a list L ′ r,s , for some r and s. A semi-visible edge is an edge of G that has one end-vertex in a list L ′ p,q , for some p and q, and one end-vertex in a list L ′′ r,s , for some r and s. An invisible edge is an edge of G that has one end-vertex in a list L ′′ p,q , for some p and q, and one end-vertex in a list L ′′ r,s , for some r and s. Intuitively, visible, semi-visible, and invisible edges are such that both end-vertices, one end-vertex, and no end-vertex, respectively,