Abstract
In this chapter, we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs and, in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: Being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.
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- 1.
Formally, the assumption is that an edge never contains a vertex in its relative interior. In the case of a complete geometric graph, which we consider in this chapter, this implies that the vertices are in general position (i.e., that no three vertices lie on the same line).
- 2.
Note that the line l(e) avoids all vertices of G except the endpoints of e, since V (G) is in general position.
- 3.
Note that usually the term “terminal edges of the spine” of a caterpillar is not defined uniquely. Here and in the sequel, we mean that there exists a spine whose terminal edges are boundary edges, and in all proofs where we consider the spine of B, we refer to a particular spine whose terminal edges are boundary edges.
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Keller, C., Perles, M.A., Rivera-Campo, E., Urrutia-Galicia, V. (2013). Blockers for Noncrossing Spanning Trees in Complete Geometric Graphs. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_20
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DOI: https://doi.org/10.1007/978-1-4614-0110-0_20
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