Transforming Stacks into Queues: Mixed and Separated Layouts of Graphs

TL;DR

This work investigates how queue numbers relate to stack and mixed numbers in linear graph layouts, with a focus on separated layouts for bipartite graphs. By developing subdivision-based reductions and tree- and grid-structure representations, the authors establish equivalences showing that bounding queue number by stack or mixed numbers is interdependent, and show that it suffices to analyze the simplest nontrivial cases, such as $1$-stack $1$-queue graphs, in both separated and non-separated settings. They prove key results like $ar{sn}(G)=ar{qn}(G)$ for bipartite graphs, provide explicit subdivision bounds (e.g., a $3$-stack subdivision with $O( ext{log}( ext{max}(s,q)))$ division vertices per edge), and demonstrate that separated $1$-stack $1$-queue graphs have constant queue number, while also outlining challenging instances that complicate transformations from mixed to pure layouts. The paper thus advances the understanding of whether bounded stack or mixed numbers force bounded queue numbers and delineates a roadmap of equivalent formulations and reductions toward resolving these fundamental questions.

Abstract

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 (Dujmović 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.

Figures (6)

• Figure 1: Linear layouts of $K_6$ verifying that $\mathop{\mathrm{sn}}\nolimits(K_6) \geq 3$ due to a $3$-twist (left), $\mathop{\mathrm{qn}}\nolimits(K_6) \geq 3$ due to a $3$-rainbow (middle), and $\mathop{\mathrm{mn}}\nolimits(K_6) \leq 2$ due to a $1$-stack $1$-queue layout (right).
• Figure 2: Various representations of a separated $1$-stack $1$-queue bipartite graph.
• Figure 3: An overview of the new and known relationships between different linear layouts.
• Figure 4: (a) Illustrating \ref{['lm:321']} (1) and its tightness for $k=2$(b). Edge colors illustrate the partition of the edges with respect to $V_1$ (black) and $V_2$ (white). The black lines signify the orders within $V_1$ (dotted) and $V_2$ (dashed).
• Figure 8: Transforming a separated mixed layout of a grid with diagonals (a) into a separated pure layout (d) by halving columns and rows (b) and reversing every second row and column (c). The blue edge color of the transformed stack edges is preserved for clarity.

Theorems & Definitions (16)

• Theorem 4
• Theorem 5
• Lemma 6: Riffle-Lemma
• Corollary 7
• Lemma 8: Lemma 2 in DW05
• Lemma 9: Lemma 12 in HW21
• Lemma 10: a special case of Lemma 21 in DW05
• Lemma 11: a special case of Lemma 22 in DW05
• Lemma 12
• Lemma 13