Maintaining $\mathsf{CMSO}_2$ properties on dynamic structures with bounded feedback vertex number

TL;DR

This paper introduces a dynamic data structure for maintaining $ ext{CMSO}_2$ properties on graphs with bounded feedback vertex number, enabling efficient updates and cycle packings, with applications to relational structures.

Contribution

It presents the first dynamic data structure for $ ext{CMSO}_2$ properties on graphs with bounded feedback vertex number, achieving logarithmic amortized update time.

Findings

Maintains $ ext{CMSO}_2$ properties with $ ext{O}_{ ext{phi,k}}( ext{log} n)$ amortized time.

Supports dynamic detection of vertex-disjoint cycle packings with $ ext{O}_k( ext{log} n)$ update time.

Extends to relational structures over binary signatures.

Abstract

Let $\varphi$ be a sentence of $\mathsf{CMSO}_2$ (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 $\varphi$ 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 ${\cal O}_{\varphi,k}(\log n)$. If we additionally assume that the feedback vertex number of $G$ never exceeds $k$, this update time guarantee is worst-case. By combining this result with a classic theorem of Erd\H{o}s and P\'osa, we give a fully dynamic data structure that maintains whether a graph contains a packing of $k$ vertex-disjoint cycles with amortized update time ${\cal O}_{k}(\log n)$. Our data structure also works in a larger generality of relational structures over binary signatures.