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Four new algorithms (RFTA1, RFTA2, GFGF2A, and RFTA2GE) handling the event in wireless sensor and robot networks based on the greedy-face-greedy (GFG) routing extended with auctions are proposed in this paper. In this paper, we assume that all robots are mobile, and after the event is found (reported by sensors), the goal is to allocate the task to the most suitable robot to act upon the event, using either distance or the robots' remaining energy as metrics. The proposed algorithms consist of two phases. The first phase of algorithms is based on face routing, and we introduced the parameter called search radius (SR) at the end of this first phase. Routing is considered successful if the found robot is inside SR. After that, the second phase, based on auctions, is initiated by the robot found in SR trying to find a more suitable one. In the simulations, network lifetime and communication costs are measured and used for comparison. We compare our algorithms with similar algorithms from the literature (k-SAAP and BFS) used for the task assignment. RFTA2 and RFTA2GE feature up to a seven-times-longer network lifetime with significant communication overhead reduction compared to k-SAAP and BFS. Among our algorithms, RFTA2GE features the best robot energy utilization.