\((2,C)\)-ordered path designs \(P(\nu, 3,1)\)

Let \(C\) be the underlying graph of a configuration of \(l\) blocks in a path design of order \(v\) and block size \(3\), \((V, \mathcal{B})\). We say that \((V, \mathcal{B})\) is \((l,C)\)-ordered if it is possible to order its blocks in such a way that each set of \(l\) consecutive blocks has the same underlying graph \(C\). In this paper, we completely solve the problem of the existence of a \((2,C)\)-ordered path design \(P(v, 3, 1)\) for any configuration having two blocks.