Discrete Mathematics and Combinatorics Commons^™

Subsets Of Vertices Of The Same Size And The Same Maximum Distance, Maria Axenovich, Dominik Duerrschnabel

Theory and Applications of Graphs

For a simple connected graph G=(V,E) and a subset X of its vertices, let

d*(X) = max {dist}_G(x,y): x,y ∈ X}

and let h*(G) be the largest k such that there are disjoint vertex subsets A and B of G, each of size k such that d*(A) = d*(B). Let h*(n) = min {h*(G): |V(G)|=n}. We prove that h*(n) =⌊(n+1)/3 ⌋, for n ≥ 6. This solves the homometric set problem restricted to the largest distance exactly. In addition we compare h*(G) with a respective function h_diam(G), where …