Eccentricity terrain of δ-hyperbolic graphs

A graph G=(V,E) is δ-hyperbolic if for any four vertices u,v,w,x, the two larger of the three distance sums d(u,v)+d(w,x), d(u,w)+d(v,x), and d(u,x)+d(v,w) differ by at most 2δ≥ 0. Recent work shows that many real-world graphs have small hyperbolicity δ. This paper describes the eccentricity terrain of a δ-hyperbolic graph. The eccentricity function e_G(v)=max{d(v,u) : u ∈ V} partitions the vertex set of G into eccentricity layers C_k(G) = {v ∈ V : e(v)=rad(G)+k}, k ∈N, where rad(G)=min{e_G(v): v∈ V} is the radius of G. The paper studies the eccentricity layers of vertices along shortest paths, identifying such terrain features as hills, plains, valleys, terraces, and plateaus. It introduces the notion of β-pseudoconvexity, which implies Gromov's ϵ-quasiconvexity, and illustrates the abundance of pseudoconvex sets in δ-hyperbolic graphs. In particular, it shows that all sets C_≤ k(G)={v∈ V : e_G(v) ≤ rad(G) + k}, k∈N, are (2δ-1)-pseudoconvex. Additionally, several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities. An O(δ |E|) time eccentricity approximation ê(v), for all v∈ V, is presented that uses distances to two mutually distant vertices and satisfies e_G(v)-2δ≤ê(v) ≤e_G(v). It also shows existence of two eccentricity approximating spanning trees T, one constructible in O(δ |E|) time and the other in O(|E|) time, which satisfy e_G(v) ≤ e_T(v) ≤e_G(v)+4δ+1 and e_G(v) ≤ e_T(v) ≤e_G(v)+6δ, respectively. Thus, the eccentricity terrain of a tree gives a good approximation (up-to an additive error O(δ)) of the eccentricity terrain of a δ-hyperbolic graph.

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