The 3-admissibility of a graph is a promising measure to identify real-world networks that have an algorithmically favourable structure. We design an algorithm that decides whether the 3-admissibility of an input graph G is at most p in time  \(O(m p^7)\) and space  \(O(n p^3)\) , where m is the number of edges in G and n the number of vertices. To the best of our knowledge, this is the first explicit algorithm to compute the 3-admissibility. The linear dependence on the input size in both time and space complexity, coupled with an ‘optimistic’ design philosophy for the algorithm itself, makes this algorithm practicable, as we demonstrate with an experimental evaluation on a corpus of 217 real-world networks. Our experimental results show, surprisingly, that the 3-admissibility of most real-world networks is not much larger than the 2-admissibility, despite the fact that the former has better algorithmic properties than the latter.