In this paper we introduce a novel graphical image representation comprising a single curve — the one-liner.The first step involves the detection and linking of image edges. We use a new technique, so-calledf "edge exploration," to simultaneously perform both tasks. This process is based on "image flows." It uses a gradient vector field and a new operator to explore image edges. Estimating the derivatives of the image is performed by using local Taylor expansions in conjunction with a weighted least-squares estimation method. This process finds all the possible image edges without any pruning, and collects information that allows us to prioritize the found edges. This enables us to select the most important edges, that form a "skeleton" of the sought representation. The next step connects the selected edges into one continuous curve — the one-liner. It orders the selected edges and finds curves connecting between them. We solve these two problems separately. Since the abstract graph setting of the first problem is NP-complete, we reduce it to a variant of TSP and compute an approximate solution to it. We solve the second problem by using Dijkstra?s shortest-path algorithm. We have a full software implementation for the entire one-liner etermination process.