Michael J. Pelsmajer

Parameterized Algorithms for Feedback Vertex Set

We present an algorithm for the parameterized feedback vertex set problem that runs in time \(O((2\lg{k}+ 2\lg{\lg{k}+ 18})^k n^2)\). This improves the previous \(O(max\{12^k, (4\lg{k})^k\}n^{\omega})\) 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 combinatorial tools and employing results from extremal graph theory. We also show that for several special classes of graphs the feedback vertex set problem can be solved in time c k n O(1) for some constant c. This includes, for example, graphs of genus \(O(\lg{n})\).

Removing even crossings

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 Toth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowskis theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

On the induced matching problem

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 do not contain an induced matching of size (n+10)/27. We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation. For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40k that is computable in linear time; this significantly improves the results of Moser and Sikdar (2007). We also show that we can decide in time O(91^k+n) whether a planar graph contains an induced matching of size at least k.

Crossing number of graphs with rotation systems

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 Hlinenys result, that computing the crossing number of a cubic graph (without rotation system) is NP-complete. We also investigate the special case of multigraphs with rotation systems on a fixed number k of vertices. For k = 1 and k = 2 the crossing number can be computed in polynomial time and approximated to within a factor of 2 in linear time. For larger k we show how to approximate the crossing number to within a factor of (k+4 4)/5 in time O(mk+2) on a graph with m edges.

Odd crossing number is not crossing number

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems).

Hanani-Tutte, Monotone Drawings, and Level-Planarity ∗

A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Toth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even number of times, then the graph has an x-monotone embedding in which the x-coordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly. This answers a question posed by Pach and Toth. We show that a further strengthening to a “removing even crossings” lemma is impossible by separating monotone versions of the crossing and the odd crossing number.

Odd Crossing Number and Crossing Number Are Not the Same

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems).

Strong Hanani-Tutte on the Projective Plane

If a graph can be drawn in the projective plane so that every two nonadjacent edges cross an even number of times, then the graph can be embedded in the projective plane.

Removing Independently Even Crossings

We show that

Crossing Numbers of Graphs with Rotation Systems

\mathrm{cr}(G)\leq({2\,\mathrm{iocr}(G)\atop2})