let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

We first devise moderately exponential exact algorithms for max k-vertex cover, with time-complexity exponential in n but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max k-vertex cover with complexity bounded above by the maximum among c and γ , for some γ < 2, whe...

This paper presents an O(1.2738 + kn)-time polynomialspace parameterized algorithm for Vertex Cover improving the previous O(1.286+kn)-time polynomial-space upper bound by Chen, Kanj, and Jia. The algorithm also improves the O(1.2745k+kn)-time exponentialspace upper bound for the problem by Chandran and Grandoni.

In this paper we de1ne the vertex-cover polynomial (G; ) for a graph G. The coe2cient of r in this polynomial is the number of vertex covers V ′ of G with |V ′|= r. We develop a method to calculate (G; ). Motivated by a problem in biological systematics, we also consider the mappings f from {1; 2; : : : ; m} into the vertex set V (G) of a graph G, subject to f−1(x) ∪ f−1(y) = ∅ for every edge x...

A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. In this paper, we give a kernel of 5k vertices and an O(1.7485)-time polynomial-space algorithm for this problem, both new results improve previous known b...

Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

the topological index of a graph g is a numeric quantity related to g which is invariant underautomorphisms of g. the vertex pi polynomial is defined as piv (g)  euv nu (e)  nv (e).then omega polynomial (g,x) for counting qoc strips in g is defined as (g,x) =cm(g,c)xc with m(g,c) being the number of strips of length c. in this paper, a new infiniteclass of fullerenes is constructed. the ...

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing wit...

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).