**MSC:**- 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
- 68R10 Graph theory, See also {05Cxx, 90B10, 90B35, 90C35}
- 90C35 Programming involving graphs or networks

union of paths and for which there is a bijective mapping $f: V \rightarrow

\{1,\ldots,|V|\}$ such that $\max\{|f(v)-f(u)|: vw \in E\}= 2$.

Results on the structure of such graphs are presented. The corresponding

properties lead to the definition of weak bandwidth--2 graphs. This more

general class can be recognized by a modification of the well--known

DFS--algorithm for the construction of the block--cutvertex graph. In this

larger class the bandwidth--2 graphs can be filtered out by a simple

procedure on the blocks and a greedy procedure which locally depends on

several cases given by the position and the length of the hairs.

Altogether, a new linear--time algorithm for the recognition of bw--2

graphs is given.