New Bounds for the Signless Laplacian Spread

Let $G$ be a simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant

Improved Exact Exponential Algorithms for Vertex Bipartization and Other Problems

1 We study efficient exact algorithms for several problems including VERTEX BIPARTIZATION, FEEDBACK VERTEX SET, 4-HITTING SET, MAX CUT in graphs with maximum degree at most 4. Our main results include: • an O ∗ (1.9526 n) 2 algorithm for VERTEX BIPARTIZATION problem in undirected graphs; • an O ∗ (1.8384 n) algorithm for VERTEX BIPARTIZATION problem in undirected graphs of maximum degree 3; • an O ∗ (1.945 n) algorithm for FEEDBACK VERTEX SET and VERTEX BIPARTIZATION problem in undirected graphs of maximum degree 4; • an O ∗ (1.9799 n) algorithm for 4-HITTING SET problem; • an O ∗ (1.5541 m) algorithm for FEEDBACK ARC SET in tournaments. To the best of our knowledge, these are the best known exact algorithms for these problems. In fact, these are the first exact algorithms for these problems with the base of the exponent < 2. En route to these algorithms, we introduce two general techniques for obtaining exact algorithms. One is through parameterized complexity algorithms, and the other is a ‘colored ’ branch-and-bound technique. I

On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of OCT with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Odd Cycle Transversal admits a polynomial kernel when parameterized by l. On the positive side, we show a polynomial kernel for OCT when parameterized by the vertex deletion distance to the class of bipartite graphs of treewidth at most w (for any constant w); this generalizes the parameter feedback vertex set number (i.e., the distance to a forest). Complementing this, we exclude polynomial kernels for OCT parameterized by the distance to outerplanar graphs, conditioned on the assumption that NP \not \subseteq coNP/poly. Thus the bipartiteness requirement for the treewidth w graphs is necessary. Further lower bounds are given for parameterization by distance from cluster and co-cluster graphs respectively, as well as for Weighted OCT parameterized by the vertex cover number (i.e., the distance from an independent set).Comment: Accepted to IPEC 2011, Saarbrucke