An Inductive Construction of (2,1)-tight Graphs

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.