Dominating Sets Whose Closed Stars Form Spanning Trees

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. AMS Subject Classification (1991): Primary: 05C35 Secondary: 05C05, 05C45, 05C85, 68R10, 90B12 1. Introduction In this paper we introduce a new variation on domination in graphs. The motivation for this research grew not from the wealth of results on dominating sets---s..