A Note on Distance Approximating Trees in Graphs

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