Graphs of treewidth at most two are the ones excluding the clique with four vertices as a minor. Equivalently, they are the graphs whose biconnected components are series-parallel.

We turn those graphs into a free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two.

First we propose a syntax for denoting them: in addition to series and parallel compositions, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equational presentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph.