Transforming Stacks into Queues: Mixed and Separated Layouts of Graphs

Some of the most important open problems for linear layouts of graphs ask for the relation between a graph's queue number and its stack number or mixed number. In such, we seek a vertex order and edge partition of $G$ into parts with pairwise non-crossing edges (a stack) or with pairwise non-nesting edges (a queue). Allowing only stacks, only queues, or both, the minimum number of required parts is the graph's stack number $sn(G)$, queue number $qn(G)$, and mixed number $mn(G)$, respectively. Already in 1992, Heath and Rosenberg asked whether $qn(G)$ is bounded in terms of $sn(G)$, that is, whether stacks "can be transformed into" queues. This is equivalent to bipartite $3$-stack graphs having bounded queue number (Dujmovi\'c and Wood, 2005). Recently, Alam et al. asked whether $qn(G)$ is bounded in terms of $mn(G)$, which we show to also be equivalent to the previous questions. We approach the problem by considering separated linear layouts of bipartite graphs. In this natural setting all vertices of one part must precede all vertices of the other part. Separated stack and queue numbers coincide, and for fixed vertex orders, graphs with bounded separated stack/queue number can be characterized and efficiently recognized, whereas the separated mixed layouts are more challenging. In this work, we thoroughly investigate the relationship between separated and non-separated, mixed and pure linear layouts.