We prove that the class of 3D-grids cannot be transduced from planar graphs, and more generally, from any class of graphs of bounded genus. To prove our result, we introduce a new structural tool called slice decompositions and study its properties. We show that every graph class transducible from a class of graphs of bounded genus is a perturbation of a graph class that admits slice decompositions. Moreover, we show that edge-stable graph classes that admit slice decomposition are transducible from weakly sparse graph classes that admits slice decompositions.

Abstract Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in $${\mathbb R}^2$$ R 2 having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most $$\omega $$ ω that the chromatic number $$\chi (G)$$ χ ( G ) of G is at most $$d\omega $$ d ω . We show for every even value of $$\omega $$ ω how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the $$\chi $$ χ -binding function of d-DIR is $$\omega \mapsto d\omega $$ ω ↦ d ω for $$\omega $$ ω even and $$\omega \mapsto d(\omega -1)+1$$ ω ↦ d ( ω - 1 ) + 1 for $$\omega $$ ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case $$d=2$$ d = 2 .

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula $F$ along with a bound $k$ and asks for the weighted sum of all models with at most $k$ positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of $F$ plus $k$. We show that this result is tight: it is neither possible to drop the bound $k$ nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures – culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking on nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking on dense classes of graphs or other structures. We study the FO model checking problem on dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarský et al. for FO model checking on posets of bounded width. We also complement the tractability results by related hardness reductions.