We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph \(G=(V,E)\) maximum edge size f, cost function \(w: V\rightarrow {\mathbb {Z}}^+\), and subsets \(P_1,P_2,\ldots ,P_r\) of E along covering requirements \(k_1,k_2,\ldots ,k_r\) for each subset. The objective is to find minimum subset S V such that, \(P_i\), at least...

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality . Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1, 2]-factor FG, i.e. a spa...

Let G be a k-connected graph of order n. In [1], Bondy considered a degree sum condition for a graph to have a hamiltonian cycle, say, to be covered by one cycle. He proved that if σk+1(G) > (k+1)(n− 1)/2, then G has a hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. [4] proved that if k = 1 and σ3(G) ≥ n, then G can...

We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, an...

Combining the concept of a fractional ( g , f )-covered graph with that ID-( )-factor-critical graph, we define covered graph. This paper reveals relationship between some parameters and existence graphs. A sufficient condition for being is presented. In addition, demonstrate sharpness main result in this by constructing special class. Furthermore, other parameters(such as binding number, tough...

A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to be NP-hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural characterization of well-covered graphs and then apply it to the recognition problem. Apparently, p...

A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. A matching covered graph G is near-bipartite if it is nonbipartite and there is a removable double ear R such that G−R is matching covered and bipartite. In 2000, C. Little and I. Fischer characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [3]. However, their charac...

A graph G is well covered, if any two maximal independent sets of G have the same number of vertices. A graph is called a block-cactus graph if each block is complete or a cycle. In this paper we characterize the well covered block-cactus graphs. 1. TERMINOLOGY AND INTRODUCTION In this paper we consider finite, undirected, and simple graphs G with the vertex set V(G). The degree d(x, G) of a ve...

We study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs.