  The simple graphs $G=(V,E)$ that satisfy $|E'|\leq 2|V'|-l$ for any subgraph
(and for $l=1,2,3$) are the $(2,l)$-sparse graphs. Those that also satisfy
$|E|=2|V|-l$ are the $(2,l)$-tight graphs. These can be characterised by their
decompositions into two edge disjoint spanning subgraphs of various types. The
Henneberg--Laman theorem characterises $(2,3)$-tight graphs inductively in
terms of two simple moves, known as the Henneberg moves. Recently this has been
extended, via the addition of a graph extension move, to the case of
$(2,2)$-tight graphs. Here an alternative characterisation is provided by means
of vertex-to-$K_4$ and edge-to-$K_3$ moves, and this is extended to the
$(2,1)$-tight graphs by addition of an edge joining move. Similar
characterisations of $(2,l)$-sparse graphs are also provided.
