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- Marcus Schaefer
- Arch. Math. Log.
- 1998

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)

- Marcus Schaefer, Daniel Stefankovic
- Theory of Computing Systems
- 2015

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)

- Marcus Schaefer, Eric Sedgwick, Daniel Stefankovic
- STOC
- 2002

A <i>string graph</i> is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in <b>NP</b>. The recognition problem was not known to be decidable until very recently, when two independent… (More)

- Iyad A. Kanj, Michael J. Pelsmajer, Marcus Schaefer
- IWPEC
- 2004

We present an algorithm for the parameterized feedback ver-tex set problem that runs in time O((2 lg k + 2 lg lg k + 18) k n 2). This improves the previous O(max{12 k , (4 lg k) k }n ω) algorithm by Raman et al. by roughly a 2 k factor (n w ∈ O(n 2.376) is the time needed to multiply two n × n matrices). Our results are obtained by developing new… (More)

- Marcus Schaefer, Daniel Stefankovic
- J. Comput. Syst. Sci.
- 2001

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)

- Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
- J. Comb. Theory, Ser. B
- 2007

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)

- Björn Kemper, Daniel Carl, +4 authors Gert von Bally
- Journal of biomedical optics
- 2006

Digital holographic microscopy provides new facilities for contactless and marker-free quantitative phase contrast imaging. In this work, a digital holographic microscopy method for the integral refractive index determination of living single cells in cell culture medium is presented. Further, the obtained refractive index information is applied to full… (More)

- Marcus Schaefer
- STOC
- 1999

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 Π p 2 = coNP NP and it was shown to be coNP-hard by Burr [Bur90]. We prove that ARROWING is Π p 2-complete, simultaneously settling a conjecture of… (More)

- Marcus Schaefer
- J. Graph Algorithms Appl.
- 2012

We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested earlier in the writings of Tutte and Wu.

- Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
- Graph Drawing
- 2007

We show that computing the crossing number of a graph with a given rotation system is NP-complete. This result leads to a new and much simpler proof of Hliněn´y's result, that computing the crossing number of a cubic graph (no rotation system) is NP-complete.