### On the Queue Number of Planar Graphs

Giuseppe Di Battista, Fabrizio Frati, Janos Pach
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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
more » ... 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,