Eccentricity function in distance-hereditary graphs
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A graph G = (V,E) is distance hereditary if every induced path of G is a shortest path. In this paper, we show that the eccentricity function e(v) = max{d(v, u) : u ∈ V } in any distance-hereditary graph G is almost unimodal, that is, every vertex v with e(v) > rad(G) + 1 has a neighbor with smaller eccentricity. Here, rad(G) = min{e(v) : v ∈ V } is the radius of graph G. Moreover, we use this result to characterize the centers of distance-hereditary graphs and provide a linear time algorithm to find a large subset of central vertices, and in some cases, all central vertices. We introduce two new algorithmic techniques to approximate all eccentricities in distance-hereditary graphs, including a linear time additive 1-approximation.
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