Improved product structure for graphs on surfaces

Abstract

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.

The product structure of squaregraphs

“…They have been the key tool to resolve several open problems regarding queue layouts [15], nonrepetitive colourings [13], centred colourings [9], clustered colourings [14], adjacency labellings [5, 16, 17], vertex rankings [7], twin‐width [6] and infinite graphs [22]. Similar product structure theorems are known for other classes, including graphs with bounded Euler genus [11, 15], apex‐minor‐free graphs [15], $$(g,d)$$‐map graphs [12], $$(g,\delta )$$‐string graphs [12], $$(g,k)$$‐planar graphs [12], powers of planar graphs [12, 20], fan‐planar graphs [20] and $$k$$‐fan‐bundle planar graphs [20].…”

Product structure of graph classes with bounded treewidth

“…The building blocks typically have bounded treewidth, which is the standard measure of how similar a graph is to a tree. Examples of graph classes that can be described this way include planar graphs [29,73], graphs of bounded Euler genus [23,29], graphs excluding a fixed minor [29], and various non-minor-closed classes [31,41]. These results have been the key to solving several open problems regarding queue layouts [29], nonrepetitive colouring [28], p-centered colouring [25], adjacency labelling [27,36], twin-width [3,11], and comparable box dimension [33].…”

Graph Colouring and Decomposition

“…The upper bound in Theorem 2.6 for bounded genus graphs follows from Theorem 2.4, Lemma 2.5, observation 2.6 and the following recent result of Distel, Hickingbotham, Huynh, and Wood [19]: 19]). For every n-vertex graph G of Euler genus at most g, there exists some at most n-vertex simple 3-tree H and some path P such that G is isomorphic to a subgraph of H ⊠ K max{2g,3} ⊠ P…”

“…Theorem 1.10 ( [46,74]). For any graph G, χ cen (G) ≤ (χ lin (G)) 19 • (log(χ lin (G))) O (1) . Kun et al [46] give a family of graphs that contains, for every ϵ > 0, a graph G with χ cen (G) ≥ (2 − ϵ) χ lin (G).…”