For a family of graphs F , a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F . F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G = (V,∪i=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colo...

Let G be a simple graph of order n and size m. An edge covering of the graph G is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Let e(G, k) be the number of edge covering sets of G of size k. The edge cover polynomial of G is the polynomial E(G, x) = m

The VERTEX COVER problem asks, for input consisting of a graph G on n vertices, and a positive integer k, whether there is a set of k vertices such that every edge of G is incident with at least one of these vertices. We give an algorithm for this problem that runs in time O(kn + (1.324718)'k'). In particular, this gives an 0((1.324718) " n2) algorithm to find the minimum vertex cover in the gr...

Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in [3], f...

We study the kernelization complexity of structural parameterizations Vertex Cover problem. Here, goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ problem an equivalent one, whose size polynomial in pre-determined parameter $G$. A long line previous research deals with based on number vertex deletions needed $G$ member simple graph class $\mathcal{F...

A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs some problems that are NP-complete on general graphs, such as the maximum...

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al. [1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not appro...

it is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (castelnuovo-mumford) regularity. as a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph g are equal provided g is well-covered bipa...

We initiate the algorithmic study of the following “structured augmentation” question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping φ : V (H) → V (G) if every edge uv of F is either an edge of G, or φ−1(u)φ−1(v) is an edge of H. Thus F contain...

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. ([20], [1]). We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so called strong crown reductions, suitable for find...