On cycle transversals and their connected variants in the absence of a small linear forest.

A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time for (sP1+P3) -free graphs for every integer s≥1 . We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. For the latter two problems we also prove that they are polynomial-time solvable for cographs; this was known already for Feedback Vertex Set and Odd Cycle Transversal

Connected vertex cover for (sP1+P5)-free graphs

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most k that induces a connected subgraph of G. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for H-free graphs if H is not a linear forest. On the other hand, the problem is known to be polynomial-time solvable for s P2-free graphs for any integer s ≥ 1. We give a polynomial-time algorithm to solve the problem for (s P1 + P5)-free graphs for every integer s ≥ 0. Our algorithm can also be used for the Weighted Connected Vertex Cover problem

Understanding the chemical mechanisms of life

Understanding the biological, medical and ecological pathways of complex transformations will be greatly aided by the chemist's molecular approach. The 2nd European Conference on Chemistry for Life Sciences brought together these diverse disciplines to consider where we are and where we will go

Minimal separators in graph classes defined by small forbidden induced subgraphs

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on sets of forbidden induced subgraphs with at most four vertices and obtain an almost complete dichotomy, leaving open only two cases