In a quest to thoroughly understand the first-order transduction hierarchy of hereditary graph classes, some questions in particular stand out; such as, what properties hold for graph classes that are first-order transductions of planar graphs (and of similar classes)? When addressing this (so-far wide open) question, we turn to the concept of a product structure – being a subgraph of the strong product of a path and a graph of bounded tree-width, introduced by Dujmović et al. [JACM 2020]. Namely, we prove that any graph class which is a first-order transduction of a class admitting such product structure, up to perturbations also meets a structural description generalizing the concept of a product structure in a dense hereditary way—the latter concept being introduced just recently by authors under the name of $\mathcal{H}$-clique-width [MFCS 2024].Using this characterization, we show that the class of the 3D grids, as well as a class of certain modifications of 2D grids, are not first-order transducible from classes admitting a product structure, and in particular not from the class of planar graphs.

Abstract. Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020] and has interesting applications in the areas of logic on graphs and in parameterized algorithmics. Very briefly, the essence of twin-width is in a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference in the neighborhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. While for many natural graph classes, it is known that their twin-width is bounded, and published upper bounds on the twin-width in nontrivial cases are very often “astronomically large,” We focus on planar graphs, which are known to already have bounded twin-width since its introduction, but it took some time for the first explicit “nonastronomical” upper bounds to come. Namely, in the order of preprint appearance, the bound was at most 183 by Jacob and Pilipczuk [arXiv, January 2022], and 583 by Bonnet, Kwon and Wood [arXiv, February 2022]. Subsequent arXiv manuscripts in 2022 improved the bound down to 37 (Bekos et al.) and 11 and 9 (both by Hliněný). We further elaborate on the approach used in the latter manuscripts, proving that the twin-width of every planar graph is at most 8 and construct a witnessing contraction sequence in linear time. Note that the currently best lower-bound planar example is of twin-width 7 by Král’ and Lamaison [arXiv, September 2022]. We also prove small explicit upper bounds on the twin-width of bipartite planar and 1-planar graphs (6 and 16) and of map graphs (38). The common denominator of all these results is the use of a novel specially crafted recursive decomposition of planar graphs, which may be found useful also in other areas.

We introduce H-clique-width, a new structural measure of graphs that aims to provide a hereditary analogue of the traditional graph product structure. The definition naturally generalises the ordinary clique-width concept. As a result, for a class H of graphs (such as the class of paths), the H-clique-width of a graph G equals the least integer t such that G is isomorphic to an induced subgraph of the strong product of a graph from H and a graph of clique-width t. We study basic properties of H-clique-width and compare it to other established structural parameters of graphs. Notably, we prove that the celebrated Planar graph product structure theorem by Dujmovic et al., and related graph product structure results, can all be formulated with the induced subgraph containment relation. In particular, every planar graph is isomorphic to an induced subgraph of the strong product of a path and a graph of tree-width 39.

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows us to solve many otherwise hard problems efficiently. Graph classes of bounded twin-width, in which appropriate contraction sequences are efficiently constructible, are thus of interest in combinatorics and in computer science. However, we currently do not know in general how to obtain a witnessing contraction sequence of low width efficiently, and published upper bounds on the twin-width in non-trivial cases are often "astronomically large". We focus on planar graphs, which are known to have bounded twin-width (already since the introduction of twin-width), but the first explicit "non-astronomical" upper bounds on the twin-width of planar graphs appeared just a year ago; namely the bound of at most 183 by Jacob and Pilipczuk [arXiv, January 2022], and 583 by Bonnet, Kwon and Wood [arXiv, February 2022]. Subsequent arXiv manuscripts in 2022 improved the bound down to 37 (Bekos et al.), 11 and 9 (both by Hliněný). We further elaborate on the approach used in the latter manuscripts, proving that the twin-width of every planar graph is at most 8, and construct a witnessing contraction sequence in linear time. Note that the currently best lower-bound planar example is of twin-width 7, by Král' and Lamaison [arXiv, September 2022]. We also prove that the twin-width of every bipartite planar graph is at most 6, and again construct a witnessing contraction sequence in linear time.