Computational Power of A Single Oblivious Mobile Agent in Two-Edge-Connected Graphs

We investigate the computational power of a single mobile agent in an $n$-node graph with storage (i.e., node memory). It has been shown that the system with one-bit agent memory and $O(1)$-bit storage is as powerful as the one with $O(n)$-bit agent memory and $O(1)$-bit storage, and thus we focus on the difference between one-bit memory agents and oblivious (i.e. zero-bit memory) agents. While it has been also shown that their computational powers are not equivalent, all the known results exhibiting such a difference rely on the fact that oblivious agents cannot transfer any information from one side to the other side across the bridge edge. Then our main question is stated as follows: Are the computational powers of one-bit memory agents and oblivious agents equivalent in 2-edge-connected graphs or not? The main contribution of this paper is to answer this question positively under the relaxed assumption that each node has $O(\log\Delta)$-bit storage ($\Delta$ is the maximum degree of the graph). We present an algorithm of simulating any algorithm for a single one-bit memory agent by one oblivious agent with $O(n^2)$-time overhead per round. Our result implies that the topological structure of graphs differentiating the computational powers of oblivious and non-oblivious agents is completely characterized by the existence of bridge edges.