A Note on Distance Approximating Trees in Graphs

Let G = (V, E) be a connected graph endowed with the standard graph-metric dG and in which longest induced simple cycle has length λ(G). We prove that there exists a tree T = (V, F) such that |dG (u, v) − dT (u, v)| ≤ ⌊λ(G) 2 ⌋ + α for all verticesu, v ∈ V, whereα = 1 if λ(G) = 4, 5 andα = 2 otherwise. The case λ(G) = 3 (i.e., G is a chordal graph) has been considered in Brandst ädt, Chepoi, and Dragan, (1999) J.Algorithms 30. The proof contains an efficient algorithm for determining such a tree T .