Geodesic Convexity in Graphs by Ignacio M. Pelayo

Author:Ignacio M. Pelayo
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Proof.

Consider a vertex x of G. Suppose that x is not a contour vertex, i.e., x is a vertex of V (G) ∖ Ct(G). Since the eccentricities of two adjacent vertices differ by at most one unit, if x is not a contour vertex, then there exists a vertex y ∈ V (G), adjacent to x, such that its eccentricity satisfies . This fact implies the existence of a shortest x 0 − x r path such that x = x 0, x i  ∉ Ct(G) for , x r  ∈ Ct(G), and for , where l = e(x).

Fig. 4.2 Ψ is a shortest z − x r path

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