Marc Hellmuth

In this contribution the computational complexity of recognizing S-composite and S-prime graphs is considered. Klav\v{z}ar et al. [Discr. Math. 244: 223-230 (2002)] proved that a graph is S-composite if and only if it admits a nontrivial path-k-coloring. The problem of determining whether there exists a path-2-coloring for a given graph is shown to be NP-complete, which in turn is utilized to show that determining whether a graph is S-composite is NP-complete and thus, determining whether a graph is S-prime is CoNP-complete. A plenty of other problems are shown to be NP-hard, using the latter results.

This paper is concerned with the fast computation of a relation $\R$ on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of $\R$ is the relation $\delta^\ast$, whose convex closure yields the product relation $\sigma$ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of $\R$ so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of $\delta^\ast$ for graphs with maximum bounded degree. Furthermore, we define \emph{quasi Cartesian products} as graphs with non-trivial $\d...

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.

The Cartesianproduct,G �ë+��RI�JUDSKV�*�DQG�+�LV�WKH�JUDSK�ZLWK�YHUWH[�VHW� V(G)×V(H) and (a,x)(b,y) is edge in E(G�ë+��ZKHQHYHU�HLWKHU��DE��LQ�(�*��DQG�[ \��RU� a=b and (xy) in E(H). Every connected,graph,has a unique prime factor decomposition with respect to the Cartesian product; see [3, Theorem 4.9]. G is called prime if its unique prime factor decomposition has only one factor, that is, G itself. The implemented,algorithm,provides,the decomposition,of cartesian graph,products based,on the decomposition,with

Abstract—It is known that for two given secondary structures (defined by position of base pairings) an RNA string can easily be found that can fold into both structures. But for more than two secondary structures this is not necessarily possible. Moreover, when four or more ...

It is known that for two given secondary structures (defined by position of base pairings) an RNA string can easily be found that can fold into both structures. For more than two secondary structures this is not necessarily possible. In this paper, we introduce pseudo edges that are used to forbid that certain base pairs can bind and therefore can be used to define the properties of possible RNA secondary structures. We study the complexity of the problem to design an RNA sequence that can fold into different secondary structures each of them is described by a set of required and forbidden base pairs. We refine the NP-completeness results of Clote et al. (2005) and show an analogous NP-completeness result for the realisation problem concerning the removal of (pseudo) edges. We also present a polynomial time method for checking the realisability of extended shape graphs. Furthermore, we empirically analyse the influence of pseudo edges on the realisability for sets of random RNA sequences and for sets of aptamers.