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Book cover Geodesic Convexity in Graphs

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Abstract

For two vertices u and v of a graph G, a vertex x ∈ V (G) is said to be geodominated by the pair {u, v} if x lies on some u − v geodesic in G. The geodetic interval I G [u, v] consists of u, v together with all vertices geodominated by the pair {u, v}. If S is a set of vertices of G, then the geodetic closure I G [S] is the union of all sets I[u, v] for u, v ∈ S, i.e., it consists of S together with all vertices lying on some geodesic joining two vertices of S. When the graph G is clear from the context, I G [u, v] and I G [S] are usually replaced by I[u, v] and I[S], respectively.

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Notes

  1. 1.

    With respect to the geodesic convexity.

  2. 2.

    As in the remaining sections and chapters, unless otherwise stated, all terms, invariants and results are referred to the geodesic convexity.

  3. 3.

    Some authors [90, 132] define the Radon number to be one unit larger, i.e., as the smaller value r such that each set with at least r points admits a Radon partition.

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    Ignacio M. Pelayo

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Pelayo, I.M. (2013). Invariants. In: Geodesic Convexity in Graphs. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8699-2_2

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