(Solvability of Edge Covering) For two robots, edge covering on G is solvable if and only if G is connected and has a simple cycle of odd length.
Another contribution, is to expose the inherent difficulty of requiring the robots to coordinate to cover an edge: edge covering is unsolvable when the number of robots is at least 3 (Lemma 6).
The problems are the graph convergence (robots decide vertices that belong to the same edge) and edge covering (robots decide vertices that cover an edge).
As we will see, the extra requirement of edge covering that processes always have to cover an edge, precludes solutions for three or more robots, for any graph.
Edge covering problem (ECP) is to find an edge cover with the minimum weight in a graph.
This paper considers the edge covering problem under fuzzy environment, and formulates three models which are expected minimum weight edge cover model, α-minimum weight edge cover model, and the most minimum weight edge cover model.
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Fig. 3 Edge covering for two robots.
Finally, from Lemmas 5 and 6 and Theorem 3 in "The edge-covering problem" section, we derive a full characterization for the solvability of edge covering.
We first show that if G is disconnected, then graph convergence and edge covering are impossible.
