 Unoriented graphs are defined as for graphs with an additional unary operation \(g\) satisfying \(g(g(x)) = x\) and \(s(g(x)) = t(x)\) (whence \(s(x) = s(g(g(x))) = t(g(x)))\). The free undirected graph on \(x\) consists of \(x, s(x), t(x)\), and \(g(x)\), with the pair \(x, g(x)\) constituting the two one-way lanes of a two-lane highway between \(s(x) = t(g(x))\) and \(t(x) = s(g(x)\)). Identification of elements of undirected graphs works as for their oriented counterparts: it is only worth identifying vertices. However there is one interesting twist here: vertices can be of two kinds, those satisfying \(x = g(x)\) and those not. The latter kind of vertex is now asymmetric: one direction of the bidirectional edge is identified with its vertices while the other one forms an oriented loop in the sense that its other direction is a vertex. This phenomenon does not arise for undirected graphs defined as those satisfying “if \(s(x) = s(y)\) and \(t(x) = t(y)\) then \(x = y\).”