In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are drawn. An abstract rotation system of a graph assigns to each vertex a cyclic order of its incident edges. A realizable rotation system is one that admits a simple drawing such that at each vertex, the edges emanate in that cyclic order, and a generalized twisted rotation system can be realized as a generalized twisted drawing. Generalized twisted drawings have initially been introduced to obtain improved bounds on the size of plane substructures in any simple drawing of K_n. They have since gained independent interest due to their surprising properties. However, the definition of generalized twisted drawings is very geometric and drawing-specific. In this paper, we develop characterizations of generalized twisted drawings that enable a purely combinatorial view on these drawings and lead to efficient recognition algorithms. Concretely, we show that for any n ≥ 7, an abstract rotation system of K_n is generalized twisted if and only if all subrotation systems induced by five vertices are generalized twisted. This implies a drawing-independent and concise characterization of generalized twistedness. Besides, the result yields a simple O(n⁵)-time algorithm to decide whether an abstract rotation system is generalized twisted and sheds new light on the structural features of simple drawings. We further develop a characterization via the rotations of a pair of vertices in a drawing, which we then use to derive an O(n²)-time algorithm to decide whether a realizable rotation system is generalized twisted.

We define a graph tile to be a unit square (or more generally, a polygon) on which a piece of a graph has been drawn/embedded; in particular, it may have vertices in its interior, edges connecting those vertices, or half-edges that extend to the boundary of the tile. In a graph tiling problem, we are given as input a set of graph tiles, with multiplicities, and the output is an arrangement of those tiles forming a graph of larger area. We focus on a simple tile set: unit square tiles with a central vertex and either a half-edge or no half-edge on each side. Up to symmetry this gives us six different types. We characterize which multiplicities are compatible for sets of at most three different tiles.

A linear layout of a graph defines a total order of the vertices and partitions the edges into either stacks or queues, i.e., crossing-free and non-nested sets of edges along the order, respectively. In this work, we study defective linear layouts that allow forbidden patterns among edges of the same set. Our focus is on k-defective stack layouts and k-defective queue layouts, in which the conflict graph representing the forbidden patterns among the edges of each stack or queue has maximum degree at most k.

Abstract Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains $$\Omega (n^{\frac{1}{2}})$$ Ω ( n 1 2 ) pairwise disjoint edges and a plane cycle (and hence path) of length $$\Omega (\frac{\log n }{\log \log n})$$ Ω ( log n log log n ) . Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.