Graph decomposition for improving memoryless periodic exploration

Abstract

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 from 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 [1]] that π≤4 1/3n 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 labelings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest. © 2009 Springer Berlin Heidelberg.