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Spanning Star Trees in Regular Graphs

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Abstract

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 S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.

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References

  1. Dunbar, Jean E., Grossman, Jerrold W., Hattingh, Johannes H., Hedetniemi, Stephen T., McRae, Alice A.: On weakly connected domination in graphs. Discrete Math. (to be published)

  2. Grossman, Jerrold W.: Dominating sets whose closed stars form spanning trees. Discrete Math. (to be published)

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Correspondence to Jerrold W. Grossman.

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Grossman, J.W. Spanning Star Trees in Regular Graphs. Graphs and Combinatorics 13, 353–358 (1997). https://doi.org/10.1007/BF03353013

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  • DOI: https://doi.org/10.1007/BF03353013

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