In this paper we study properties for the structure of an undirected graph that is not vertex connected We also study the evolution of this structure when an edge is added to optimally increase the vertex connectivity of the underlying graph Several properties reported here can be extended to the case of a graph that is not k vertex connected for an arbitrary k Using properties obtained here we...

We show that there exists a polynomial algorithm to pack interval graphs with vertex-disjoint triangles.

Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles and paths, respectively. They satisfy recurrence relations with respect to elementary edge or vertex operations. They are related to other polynomials and ca...

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem Q if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we sh...

We present a novel approach for preprocessing systems of polynomial equations via graph partitioning. The variable-sharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the corresponding system of equations can be split into smaller ones that can be solved individually. This can provide a tremendous speed-up in computing the solution to the system, bu...

Let G = (V,E) be a simple undirected graph with n vertices then a set partition π = {V1, . . . , Vk} of the vertex set of G is a connected set partition if each subgraph G[Vj ] induced by the blocks Vj of π is connected for 1 ≤ j ≤ k. Define qi(G) as the number of connected set partitions in G with i blocks. The partition polynomial is Q(G,x) = ∑n i=0 qi(G)x . This paper presents a splitting ap...

TheOdd 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 Od...

We present a novel approach for solving systems of polynomialequations via graph partitioning. The concept of a variable-sharing graph of asystem of polynomial equations is defined. If such graph is disconnected, thenthe system of equations is actually two separate systems that can be solvedindividually. This can provide a significant speed-up in computing the solutionto the...

We consider the problem of computing Fréchet distance between two curves for which exact locations vertices are unknown. Each vertex may be placed in a given uncertainty region that vertex, and objective is to place so as minimise distance. This was recently shown NP-hard 2D, it unclear how compute an optimal placement at all. present first general algorithmic framework this problem. prove resu...

Abstract We consider the following problem: for a given graph G and two integers k d , can we apply fixed operation at most times in order to reduce parameter $$\pi $$ π by least ? show that this problem is NP-hard when independence number vertex deletion or edge contraction, even $$d=1$$ d...