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In this paper we explore a fundamental joint routing and scheduling problem in wireless mesh networks (WMNs) that employ time division multiple access (TDMA). The problem, referred to as the minimum cost single flow routing and scheduling (MC-SFRS) problem, deals with incremental update of transmission schedules necessitated by dynamic arrival of new flows and termination of existing flows during the operation of the network. In the problem, we are given a multi-hop WMN, a set of ongoing flows, a transmission schedule for the ongoing flows, a set of costs associated with links, and a new flow demand. All flows contend for using one of the available wireless channels. The problem asks for finding a non-bifurcated route with minimum cost along which the new flow can be scheduled without perturbing slot assignments in the given schedule, if such route exists. Our main contribution is an efficient algorithm for solving the MC-SFRS problem for arbitrary interference relations among pairs of transmission links in networks with arbitrary topologies. Among other classes of routes, our algorithm is exact over the class of shortest routes. The obtained simulation results demonstrate the effectiveness of our proposed algorithm over the competing method of exact scheduling for a fixed routing tree. In addition, the results show improvement obtained by using our algorithm to augment the schedules obtained by fixed tree routing.