Observations on Self - Stabilizing Graph

We investigate the existence of deterministic uniform self-stabilizing algorithms (DUSSAs) for a number or problems on connected undirected graphs. This investigation is carried out under three models of parallelism, namely central daemon, restricted parallelism, and maximal paral-lelism. We observe that for several problems including 2-coloring odd-degree complete bipartite graphs, 2-coloring trees, nding maximal independent sets in general graphs, and obtaining a valid coloring of planar graphs, no DUSSAs exist under the maximal parallelism model. A DUSSA for the 6-coloring problem for planar graphs under the central daemon model was presented in GK93]. For the other problems listed above, we present DUSSAs under the central daemon model. We observe that these DUSSAs work correctly under a restricted parallelism model as well. This fact enables us to apply a technique in SRR94] to obtain randomized USSAs (RUSSAs) under the maximal paral-lelism model for all the above problems. These RUSSAs achieve self-stabilization with probability 1. We also observe that techniques due to Angluin Ang80] lead to general results that establish the non-existence of DUSSAs for a large collection of graph problems under any of the parallelism models. The problems in this collection include determining the parity of the number of nodes in a graph and membership testing for various graph classes (for example, planar graphs, chordal graphs, and interval graphs).