Eva Jelínková

The planar slope number of planar partial 3-trees of bounded degree

It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope number of every series-parallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most

Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters

2^{{\mathcal O}(\Delta)}

Clustered Planarity: Small Clusters in Cycles and Eulerian Graphs

. In particular, we answer the question of Dujmovic et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f(Δ) slopes.

Clustered planarity: small clusters in Eulerian graphs

We present a polynomial-time algorithm for c-planarity testing of clustered graphs with fixed plane embedding and such that every cluster induces a subgraph with at most two connected components.

The Möbius function of separable and decomposable permutations

We present several polynomial-time algorithms for c-planarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(|C| + n)-time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graphs, namely graphs obtained from a 3-connected planar graph of fixed size k by multiplying and then subdividing edges. An O(3 · k · n)-time algorithm is presented. We further give an O(|C| + n)-time algorithm for general 3-connected planar graphs. Submitted: December 2007 Reviewed: November 2008 Revised: February 2009 Accepted: August 2009 Final: September 2009 Published: November 2009 Article type: Regular paper Communicated by: S.-H. Hong and T. Nishizeki An extended abstract of this paper appeared in proceedings of Graph Drawing 2007. [14] Department of Applied Mathematics is supported by project 1M0021620838 of the Czech Ministry of Education. Institute for Theoretical Computer Science is supported by grant 1M0545 of the Czech Ministry of Education. The 4th author was supported by the grant GAUK 154907. The 5th author and the 6th author were supported by grant 201/05/H014 of the Czech Science Foundation. The 5th author was also partially supported by the ERASMUS program and by the DFG, project NI 369/4 (PIAF), while visiting Friedrich-Schiller-Universitat Jena, Germany (October 2008–

On Switching to H-Free Graphs

We present several polynomial-time algorithms for c-planarity testing for clustered graphs with clusters of size at most three. The most general result concerns a special class of Eulerian graphs, namely graphs obtained froma fixed-size 3-connected graph bymultiplying and then subdividing edges. We further give algorithms for 3-connected graphs, and for graphs with small faces. The last result applies with no restrictions on the cluster size.

On the Hardness of Switching to a Small Number of Edges

We give a recursive formula for the Mobius function of an interval [@s,@p] in the poset of permutations ordered by pattern containment in the case where @p is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,...,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Mobius function in the case where @s and @p are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. We also show that the Mobius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Mobius function of an interval [@s,@p] of separable permutations is bounded by the number of occurrences of @s as a pattern in @p. Another consequence is that for any separable permutation @p the Mobius function of (1,@p) is either 0, 1 or -1.

Switching to hedgehog-free graphs is NP-complete

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H-free graph. Polynomial-time algorithms are known for H having at most three vertices or isomorphic to P4. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP-complete, thus solving an open problem [Kratochvil, Nesetřil and Zýka, Ann Discrete Math 51 1992]. Further, we give a characterization of graphs switching equivalent to a K1, 2-free graph by ten forbidden-induced subgraphs, each having five vertices. We also give the forbidden-induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.

Evaluation of geochemical data acquired from regular grids

Seidel’s switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches.

Aberrant splicing of the COL4A5 gene in patients with Alport syndrome

We study the problem of deciding if, for a fixed graph H, a given graph is switching-equivalent to an H-free graph. In all cases of H that have been solved so far, the problem is decidable in polynomial time. We give infinitely many graphs H such that the problem is NP-complete, thus solving an open problem [Kratochvil, Nesetřil and Zýka, Ann. Discrete Math. 51 (1992)].