  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 two similar
algorithms from the literature (k-SAAP and BFS) used for the task assignment.
RFTA2 and RFTA2GE feature up to 7 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.
