Queue Number Calculator

queue layout width

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About Queue Number Calculator

A queue number calculator computing qn(G): minimum queues for a queue layout (vertices on line, edges in FIFO queues without nesting). qn=1 iff arched-leveled-planar. Planar graphs: bounded qn (breakthrough 2020). Client-side.

Queue Number Calculator Features

  • qn(G) value
  • Queue layout
  • Planar bounded
  • FIFO vs stack
  • Common graphs
Queue number qn(G): minimum queues for queue layout. Vertices on line, edges assigned to queues. Queue constraint: no nested edges (FIFO). Planar graphs have bounded qn ≤ 49 (Dujmović-Joret-Micek-Morin-Ueckerdt-Wood, 2020). Major breakthrough!

How to Use

Select graph:

  • qn: Queue number
  • Layout: Vertex order + queues
  • Compare: qn vs bt

2020 Breakthrough

Open for 30+ years: do planar graphs have bounded queue number? Dujmović et al. proved qn(planar) ≤ 49 using layered treewidth. Extended to bounded-genus, minor-free graphs. One of the biggest results in graph drawing.

Queue vs Stack

Stack layout (book embedding): LIFO, no crossings on page. Queue layout: FIFO, no nesting. These capture different linear structures. Stack number can differ greatly from queue number for the same graph.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute qn(G).
  3. 3Find queue layout.
  4. 4Compare to bt.
  5. 5Check bounds.

Queue Number Calculator — Frequently Asked Questions

What's a queue layout?+

Vertices on a line, edges assigned to queues. Queue constraint: if edge ab and cd are in the same queue with a<c<d<b, that's 'nesting' = forbidden. FIFO order: first-in-first-out arrangement.

Why was the planar graph result important?+

Heath-Leighton-Rosenberg (1992) conjectured bounded qn for planar graphs. Open 28 years! Dujmović et al. (2020) proved qn ≤ 49. Used product structure theorem: planar ⊆ path ⊠ treewidth-3. Revolutionary technique.

What's qn(K_n)?+

qn(K_n) = ⌊n/2⌋. Same as bt(K_n)! For complete graphs, stack and queue numbers coincide. But for other graphs they can differ. Trees: bt=1 but qn=1 too.

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