Covering a hypergraph of subgraphs

AbstractLet G be a tree and let H be a collection of subgraphs of G, each having at most d connected components. Let ν(H) denote the maximum number of members of H no two of which share a common vertex, and let τ(H) denote the minimum cardinality of a set of vertices of G that intersects all members of H. It is shown that τ(H)⩽2d2ν(H). A similar, more general result is proved replacing the assumption that G is a tree by the assumption that it has a bounded tree-width. These improve and extend results of various researchers