We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through Path Discovery and Shortest Path Discovery, where the task is to realize a vertex set containing an s-t-path or a shortest s-t-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.

In the Vertex-Disjoint-Paths-With-Congestion problem, the input consists of a digraph D, an integer c and k pairs of vertices (s_i, t_i), and the task is to find a set of paths connecting each s_i to its corresponding t_i, whereas each vertex of D appears in at most c many paths. The case where c = 1 is known to be NP-complete even if k = 2 [Fortune, Hopcroft and Wyllie, 1980] on general digraphs and is W[1]-hard with respect to k (excluding the possibility of an f(k)n^O(1)-time algorithm under standard assumptions) on acyclic digraphs [Slivkins, 2010]. The proof of [Slivkins, 2010] can also be adapted to show W[1]-hardness with respect to k for every congestion c ≥ 1. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D does not contain an acyclic (5, 5)-grid as a butterfly minor, (3) D does not contain an acyclic tournament on 9 vertices as a butterfly minor, and (4) D has ear-anonymity at most 5. Further, we also show that the edge-congestion variant of the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D has maximum undirected degree 3, (3) D does not contain an acyclic (7, 7)-wall as a weak immersion and (4) D has ear-anonymity at most 5.