  Graphs have been widely used in real-world applications, in which
investigating relations between vertices is an important task. In this paper,
we study the problem of generating the k-hop-constrained s-t simple path graph,
i.e., the subgraph consisting of all simple paths from vertex s to vertex t of
length no larger than k. To our best knowledge, we are the first to formalize
this problem and prove its NP-hardness on directed graphs. To tackle this
challenging problem, we propose an efficient algorithm named EVE, which
exploits the paradigm of edge-wise examination rather than exhaustively
enumerating all paths. Powered by essential vertices appearing in all simple
paths between vertex pairs, EVE distinguishes the edges that are definitely (or
not) contained in the desired simple path graph, producing a tight upper-bound
graph in the time cost $\mathcal{O}(k^2|E|)$. Each remaining undetermined edge
is further verified to deliver the exact answer. Extensive experiments are
conducted on 15 real networks. The results show that EVE significantly
outperforms all baselines by several orders of magnitude. Moreover, by taking
EVE as a built-in block, state-of-the-art for hop-constrained simple path
enumeration can be accelerated by up to an order of magnitude.
