Graph Covering Using Bounded Size Subgraphs

A variant of graph covering problem demands to find a set of sub-graphs when the union of sub-graphs contain all the edges of G. Another variant of graph covering problem requires finding a collection of subgraphs such that the union of the vertices of subgraphs forms a vertex cover. We study the later version of the graph covering problem. The objective of these problems is to minimize the size/cost of the collection of subgraphs. Covering graphs with the help of a set of edges, set of vertices, tree or tour has been studied extensively in the past few decades. In this paper, we study a variant of the graph covering problem using two special subgraphs. The first problem is called bounded component forest cover problem. The objective is to find a collection of minimum number of edge-disjoint bounded weight trees such that the vertices of the forest, i.e., collection of edge-disjoint trees, cover the graph. The second problem is called bounded size walk cover problem. It asks to minimize the number of bounded size walks which can cover the graph. Walks allow repetition of vertices/edges. Both problems are a generalization of classical vertex cover problem, therefore, are NP-hard. We give 4. and 6. factor approximation algorithm for bounded component forest cover and bounded size walk cover problems respectively, where. is an approximation factor to find a solution to the tree cover problem.