Maintaining ๐ข๐ฌ๐ฒ๐ฎ_2 properties on dynamic structures with bounded feedback vertex number
share
Let ฯ be a sentence of ๐ข๐ฌ๐ฒ๐ฎ_2 (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether ฯ is satisfied in G. The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time O_ฯ,k(log n). By combining this result with a classic theorem of Erdลs and Pรณsa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time O_k(log n). Our data structure also works in a larger generality of relational structures over binary signatures.
READ FULL TEXT