Minimizing and Computing the Inverse Geodesic Length on Trees
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The inverse geodesic length (IGL) of a graph G=(V,E) is the sum of inverse distances between every two vertices: IGL(G) = ∑_{u,v}⊆ V1/d_G(u,v). In the MinIGL problem, the input is a graph G, an integer k, and a target inverse geodesic length T, and the question is whether there are k vertices whose deletion decreases the IGL of G to at most T. Aziz et al. (2018) proved that MinIGL is W[1]-hard for parameter treewidth, but the complexity status of the problem remains open in the case where G is a tree. We show that MinIGL on trees is subexponential by giving a 2^O((n n)^5/6) time, polynomial space algorithm, where n is the number of vertices of the input graph. The distance distribution of a graph G is a sequence {a_i} describing the number of vertex pairs distance i apart in G: a_i = |{{u, v}: d_G(u, v) = i}|. Given only the distance distribution, one can easily determine graph parameters such as diameter, Wiener index, and particularly, the IGL. We show that the distance distribution of a tree can be computed in O(n ^2 n) time by reduction to polynomial multiplication. We also extend the result to graphs of bounded treewidth by showing that the first P values of the distance distribution can be computed in 2^O(tw(G)) n^1 + ε√(P) time.
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