Tree t-Spanners of a Graph: Minimizing Maximum Distances Efficiently

Abstract

A tree t-spanner of a graph G is a spanning subtree T in which the distance between any two adjacent vertices of G is at most t. The smallest t for which G has a tree t-spanner is the tree stretch index. The problem of determining the tree stretch index has been studied by: establishing lower and upper bounds, based, for instance, on the girth value and on the minimum diameter spanning tree problem, respectively; and presenting some classes for which t is a tight value. Moreover, in 1995, the computational complexities of determining whether \(t = 2\) or \(t \ge 4\) were settled to be polynomially time solvable and NP-complete, respectively, while deciding if \(t = 3\) still remains an open problem.