  We give alternate constructions of (i) the scaling limit of the uniform
connected graphs with given fixed surplus, and (ii) the continuum random
unicellular map (CRUM) of a given genus that start with a suitably tilted
Brownian continuum random tree and make `horizontal' point identifications, at
random heights, using the local time measures. Consequently, this can be seen
as a continuum analogue of the breadth-first construction of a finite connected
graph. In particular, this yields a breadth-first construction of the scaling
limit of the critical Erd\H{o}s-R\'enyi random graph which answers a question
posed in [2]. As a consequence of this breadth-first construction we obtain
descriptions of the radii, the distance profiles, and the two point functions
of these spaces in terms of functionals of tilted Brownian excursions.
