Abstract
A k-queue layout is a special type of a linear layout, in which the linear order avoids \((k+1)\)-rainbows, i.e., \(k+1\) independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e., the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [13] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: (i) an algorithm to obtain straight-line drawings of outerplanar graphs, in which the y-distance of any two adjacent vertices is 1 or 2, (ii) an algorithm to obtain 5-queue layouts of planar 3-trees, and (iii) a decomposition of a planar graph into so-called tripods. In this work, we push further each of these techniques to obtain the first non-trivial improvement on the upper bound from 49 to \(42 \).
The work of C. Raftopoulou is funded by the National Technical University of Athens research program \(\mathrm {\Pi }\)EBE 2020.
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Bekos, M.A., Gronemann, M., Raftopoulou, C.N. (2021). On the Queue Number of Planar Graphs. In: Purchase, H.C., Rutter, I. (eds) Graph Drawing and Network Visualization. GD 2021. Lecture Notes in Computer Science(), vol 12868. Springer, Cham. https://doi.org/10.1007/978-3-030-92931-2_20
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