Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the inp...

Consider a graph and a collection of (not necessarily edge-disjoint) connected spanning subgraphs (factors) of the graph. We consider the problem of coloring the vertices of the graph so that each color class of the vertices dominates each factor. We find upper and lower bounds on α(t, k), which we define as the minimum radius of domination d such that every graph with a collection of k factors...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10, (see Brent [1]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] ...