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...

Let α(G) and β(G) be the independent number and vertex covering number of G, respectively. The Kronecker Product G1 ⊗ G2 of graph of G1 and G2 has vertex set V (G1 ⊗ G2) = V (G1) × V (G2) and edge set E(G1 ⊗ G2) = {(u1v1)(u2v2)|u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. In this paper, let G is a simple graph with order p, we prove that, α(Km,n⊗G)= max {(m+n)α(G),p max{m,n}} and β(Km,n⊗G) =min {(m + n)β(G)...

It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of S G (x, y), the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag comple...

For the class of monotone boolean functions f : {0,1}n → {0,1} where all 1-certificates have size 2, we prove the tight bound n (λ+ 2)2/4, where λ is the size of the largest 0-certificate of f . This result can be translated to graph language as follows: for every graph G= (V ,E) the inequality |V | (λ+ 2)2/4 holds, where λ is the size of the largest minimal vertex cover of G. In addition, ther...

Using known results regarding PCP, we present simple proofs of the inapproximability of vertex cover for hypergraphs. Specifically, we

An unsatisfiable set is a set of formulas whose conjunction is unsatisfiable. Every unsatisfiable set can be corrected, i.e., made satisfiable, by removing a subset of its members. The subset whose removal yields satisfiability is called a correction subset. Given an unsatisfiable set F there is a well known hitting set duality between the unsatisfiable subsets of F and the correction subsets o...

Let G be a simple graph of order n. The path cover number μ(G) is defined to be the minimum number of vertex disjoint paths required to cover the vertices of G. Ore proved that in general μ(G) ≤ max{1, n − σ2(G)}. We conjecture that if G is k-regular, then μ(G) ≤ n k+1 and we prove this for k ≤ 5.

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerpla...