Linear-Time Algorithms for Graphs of Bounded Rankwidth: A Fresh Look Using Game Theory

TL;DR

This paper offers a new, accessible game-theoretic proof that problems expressible in MSO are solvable in linear time on graphs with bounded rankwidth, avoiding complex logic background.

Contribution

It provides an alternative, self-contained proof of a key theorem using game theory, which can be extended to other graph parameters like treewidth.

Findings

Proof is self-contained and intuitive

Applicable to graphs of bounded rankwidth

Potential to generalize to other graph parameters

Abstract

We present an alternative proof of a theorem by Courcelle, Makowski and Rotics which states that problems expressible in MSO are solvable in linear time for graphs of bounded rankwidth. Our proof uses a game-theoretic approach and has the advantage of being self-contained, intuitive, and fairly easy to follow. In particular, our presentation does not assume any background in logic or automata theory. We believe that it is good to have alternative proofs of this important result. Moreover our approach can be generalized to prove other results of a similar flavor, for example, that of Courcelle's Theorem for treewidth.