In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization several results by various researchers. Let $G$ be multigraph in which no quadrilaterals share edges with triangles other let $\mu_G(v)=\max\{\mu_G(u,v):u\in V(G)\setminus\{v\}\}$, where $\mu_G(u,v)$ is number joining $u$ $v$ $G$. We show that for any two functions $a,b...

The minimum vertex cover problem (MVCP) is a well-known combinatorial optimization of graph theory. MVCP an NP (nondeterministic polynomial) complete and it has exponential growing complexity with respect to the size graph. No algorithm exits till date that can exactly solve in deterministic polynomial time scale. However, several algorithms are proposed approximately short Such useful for larg...

When a certain procedure is applied to extract two component processes from a single observed process, it is necessary to impose a set of restrictions that defines two components. One popular restriction is the assumption that the shocks to the trend and cycle are orthogonal. Another is the assumption that the trend is a pure random walk process. The unobserved components (UC) model (Harvey, 19...

We explain how to organize the atoms resulting from clique minimal separator decomposition into a metagraph which we call the atom graph, and give an efficient recursive algorithm to compute this graph at no extra cost than computing the atoms.