The Complexity of Obtaining a Distance-Balanced Graph

An unweighted, connected graph is distance-balanced (also called self-median) if there exists a number d such that, for any vertex v, the sum of the distances from v to all other vertices is d. An unweighted connected graph is strongly distancebalanced (also called distance-degree regular) if there exist numbers d1, d2, d3, . . . such that, for any vertex v, there are precisely dk vertices at distance k from v. We consider the following optimization problem: given a graph, add the minimum possible number of edges to obtain a (strongly) distance-balanced graph. We show that the problem is NP-hard for graphs of diameter three, thus answering the question posed by Jerebic et al. [Distance-balanced graphs; Ann. Comb. 2008]. In contrast, we show that the problem can be solved in polynomial time for graphs of diameter 2.