In this paper we show that the problem of identifying an edge (i, j) in a graph G such that there exists an optimal vertex cover S of G containing exactly one of the nodes i and j is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee 2 − 1 1+σ , where σ is an upper bound on a measure rela...

The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices form a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namel...

The Constrained Bipartite Vertex Cover problem asks, for a bipartite graph G with partite sets A and B, and integers kA and kB , whether there is a vertex cover for G containing at most kA vertices from A and kB vertices from B. The problem has an easy kernel with 2kA · kB edges and 4kA · kB vertices, based on the fact that every vertex in A of degree more than kB has to be included in the solu...

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H . It is known that Connected Ver...

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for $H$-free graphs if $H$ is not a linear forest (a graph is $H$-free if it does not contain $H$ as an induced subgraph). It is easy to see that Con...

It has been a challenging open problem whether there is a polynomial time approximation algorithm for the Vertex Cover problem whose approximation ratio is bounded by a constant less than 2. In this paper, we study the Vertex Cover problem on graphs with perfect matching (shortly, VC-PM). We show that if the VC-PM problem has a polynomial time approximation algorithm with approximation ratio bo...

In the vertex cover problem we are given a graph G = (V,E) and an integer k and have to determine whether there is a set X ⊆ V of size at most k such that each edge in E has at least one endpoint in X. The problem can be easily solved in time O∗(2k), making it fixed-parameter tractable (FPT) with respect to k. While the fastest known algorithm takes only time O∗(1.2738k), much stronger improvem...

Many graph problems cannot be computed in polynomial time unless P = NP , which most computer scientists and mathematicians doubt. Examples are the Traveling Salesman Problem (TSP), vertex cover, and dominating set. TSP is to find a Hamiltonian cycle with the least weight in a complete weighted graph. A vertex cover of a graph is a subset of the vertices that covers all edges. A dominating set ...

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F , the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F . Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when p...

We develop a polynomial time 3-2-approximation algorithm to solve the vertex cover problem on a class of graphs satisfying a property called “active edge hypothesis”. The algorithm also guarantees an optimal solution on specially structured graphs. Further, we give an extended algorithm which guarantees a vertex cover S1 on an arbitrary graph such that |S1| ≤ 32 |S*| + ξ where S* is an optimal ...