From Tripods to Bipods: Reducing the Queue Number of Planar Graphs Costs Just One Leg

As an alternative to previously existing planar graph product structure theorems, we prove that every planar graph $G$ is a subgraph of the strong product of $K_2$, a path and a planar subgraph of a $4$-tree. As an application, we show that the queue number of planar graphs is at most $38$ whereas the queue number of planar bipartite graphs is at most $25$.