Graph Decomposition for Memoryless Periodic Exploration

We consider a general framework in which a memoryless robot periodically explores all the nodes of a connected anonymous graph by following local information available at each vertex. For each vertex?v, the endpoints of all edges adjacent to v are assigned unique labels within the range?1 to?deg?(v) (the degree of?v). The generic exploration strategy is implemented using a right-hand-rule transition function: after entering vertex v via the edge labeled?i, the robot proceeds with its exploration, leaving via the edge having label [i mod deg?(v)]+1 at v. A lot of attention has been given to the problem of labeling the graph so as to achieve a periodic exploration having the minimum possible length?π. It has recently been proved (Czyzowicz et?al., Proc.?SIROCCO’09, 2009) that $\pi\leq4\frac{1}{3}n$ holds for all graphs of n vertices. Herein, we provide a new labeling scheme which leads to shorter exploration cycles, improving the general bound to π≤4n?2. This main result is shown to be tight with respect to the class of labellings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest.