Generalized trees related with tree metrics

Graphs without induced subgraphs of minimum degree at least 3 have been considered in matroid theory (see, e.g., Welsh 1976, ch. 14 or Aigner 1979, ch. 7) and in some domains of applications (Todd 1989, Guenoche and Leclerc 1999). Maximal such graphs were characterized in Toddʹs paper. They are called here 2d-trees. lass of 2d-trees consists of the classical 2-trees (Pippert and Beineke 1969), whose interesting status among chordal, series parallel and 2-connected graphs has been extensively studied in the literature (Arnborg and proskurowski 1989, Borie, Parker and Tovey 1991, Rose 1974, and others). We consider here valued 2d-trees and 2-trees with a given finite vertex set X.