Marcus Schaefer

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We show that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential theory of the reals. Comparing this to similar results on the rectilinear crossing number and intersection graphs of line segments, we argue that there is a need to recognize this level of complexity as its own class.
An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this(More)
The set of minimal indices of a GGdel numbering ' is deened as MIN' = fe : (8i < e))'i 6 = 'e]g. It has been known since 1972 that MIN' T ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observations including that MIN' is autoreducible, but neither(More)
In the Ramsey theory of graphs F → (G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H . ARROWING, the problem of deciding whether F → (G, H), lies in Πp2 = coNP NP and it was shown to be coNP-hard by Burr [Bur90]. We prove that ARROWING is Πp2-complete, simultaneously settling a conjecture of(More)
We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing(More)
We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a drawing realizing the string graph in the plane. This upper bound confirms a conjecture by Kratochv\'{\i}l and Matou\v{s}ek~\cite{KM91} and settles the long-standing open problem of the decidability of(More)
We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two(More)
We study extremal questions on induced matchings in certain natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of size at least n/40 while there are planar twinless graphs that(More)
We introduce the complexity class ∃ ℝ $\exists \mathbb {R}$ based on the existential theory of the reals. We show that the definition of ∃ ℝ $\exists \mathbb {R}$ is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known(More)