Dominating sets whose closed stars form spanning trees Original Research Article

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If there exists a W such that S(W) is a tree containing all the vertices of G, then S(W) is a spanning star tree of G. These and associated notions are related to connected and/or acyclic dominating sets and also arise in the study of A-trails in Eulerian plane graphs. Among the results in this paper are a characterization of those values of n and m for which there exists a connected graph with n vertices and m edges that has no spanning star tree, and a proof that finding spanning star trees is in general NP-hard.