Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. Intensive research into the Vertex Cover problem has shown that there is a preprocessing algorithm which given an instance (G, k) of Vertex Cover outputs an equivalent instance (G′, k′) in polynomial time with the guarantee that G′ has at most 2k′ vertice...

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graphG has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP ⊆ coNP/poly, Pathwidth admits no polynomial kernelization ev...

Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We prove that computing the vertex centroid of an H-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for H-polytopes. We also consider the problem of approximating the vertex centroid by findin...

A pseudoforest is a graph whose connected components have at most one cycle. Let X be a pseudoforest modulator of graph G, i. e. a vertex subset of G such that G−X is a pseudoforest. We show that Vertex Cover admits a polynomial kernel being parameterized by the size of the pseudoforest modulator. In other words, we provide a polynomial time algorithm that for an input graph G and integer k, ou...

for a graph $g$ with edge set $e(g)$, the multiplicative sum zagreb index of $g$ is defined as$pi^*(g)=pi_{uvin e(g)}[d_g(u)+d_g(v)]$, where $d_g(v)$ is the degree of vertex $v$ in $g$.in this paper, we first introduce some graph transformations that decreasethis index. in application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum zagreb ...

Abstract. An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Thresh...