In ordinary graphs the weights of edges and vertices are considered independently where the length of a path is simply the sum of weights of the edges and the vertices on this path. However, in many practical matters, weights at a vertex are not the same for all paths passing this vertex but depend on coming and leaving edges. The flow increasing path is a directed path in the residual network through from the source point a to the sink point z in such a way that we can send positive flow from point a to point z. To figure out the flow increasing path, we apply the depth-first search algorithm in the residual network on the extended network. After each time of regulating in the flow increment, its value increases by at least 1 unit. Since the flow increment is limited, the algorithm terminates after a finite number of computation steps. The purpose of proposing and creating an extended network model is to apply more accurate and effective modeling of real-world matters from the depth-first search algorithm in the residual network. The idea of the depth-first search algorithm in the residual network is developed and creates a new increasing flow path to find max-flow on the extended network which is different from the breadth-first search algorithm to find max-flow on the extended network. The main contribution of this paper is the augmenting-path maxflow algorithm and the Maxflow -Mincut theorem on extended networks.