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@inproceedings{2244009,
author = {Bekos, Michael A. and Da Lozzo, Giordano and Hliněný, Petr and Kaufmann, Michael},
address = {Dagstuhl, Germany},
booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
doi = {http://dx.doi.org/10.4230/LIPIcs.ISAAC.2022.23},
edition = {LIPIcs 248},
editor = {Bae, Sang Won and Park, Heejin},
keywords = {Graph product structure theory; h-framed graphs; k-map graphs; queue number; twin-width},
howpublished = {elektronická verze "online"},
language = {eng},
location = {Dagstuhl, Germany},
isbn = {978-3-95977-258-7},
pages = {"23:1"-"23:15"},
publisher = {Schloss Dagstuhl},
title = {Graph Product Structure for h-Framed Graphs},
url = {https://doi.org/10.4230/LIPIcs.ISAAC.2022.23},
year = {2022}
}
TY - JOUR
ID - 2244009
AU - Bekos, Michael A. - Da Lozzo, Giordano - Hliněný, Petr - Kaufmann, Michael
PY - 2022
TI - Graph Product Structure for h-Framed Graphs
PB - Schloss Dagstuhl
CY - Dagstuhl, Germany
SN - 9783959772587
KW - Graph product structure theory
KW - h-framed graphs
KW - k-map graphs
KW - queue number
KW - twin-width
UR - https://doi.org/10.4230/LIPIcs.ISAAC.2022.23
N2 - Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020]. In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3⌊ h/2 ⌋+⌊ h/3 ⌋-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ⌊ 33/2(k+3 ⌊ k/2⌋ -3)⌋ to ⌊ 33/2 (3⌊ k/2 ⌋+⌊ k/3 ⌋-1) ⌋, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.
ER -
BEKOS, Michael A., Giordano DA LOZZO, Petr HLINĚNÝ and Michael KAUFMANN. Graph Product Structure for h-Framed Graphs. Online. In Bae, Sang Won and Park, Heejin. \textit{33rd International Symposium on Algorithms and Computation (ISAAC 2022)}. LIPIcs 248. Dagstuhl, Germany: Schloss Dagstuhl, 2022, p.~''23:1''-''23:15'', 15 pp. ISBN~978-3-95977-258-7. Available from: https://dx.doi.org/10.4230/LIPIcs.ISAAC.2022.23.
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