MSOL Restricted Contractibility to Planar Graphs

TL;DR

This paper introduces a fixed-parameter tractable algorithm for a generalized graph planarization problem via edge contraction, incorporating MSOL constraints, and demonstrates NP-completeness for certain cases.

Contribution

The paper develops an FPT algorithm for MSOL-restricted edge contraction problems and extends it to specific subgraph contractibility cases, with complexity analysis.

Findings

FPT algorithm for P-RESTRICTEDCONTRACT with MSOL constraints

Solution for $ ext{ell}$-subgraph contractibility problem

NP-completeness for $ ext{ell} extgreater 1$ cases

Abstract

We study the computational complexity of graph planarization via edge contraction. The problem CONTRACT asks whether there exists a set $S$ of at most $k$ edges that when contracted produces a planar graph. We work with a more general problem called $P$-RESTRICTEDCONTRACT in which $S$, in addition, is required to satisfy a fixed MSOL formula $P(S,G)$. We give an FPT algorithm in time $O(n^2 f(k))$ which solves $P$-RESTRICTEDCONTRACT, where $P(S,G)$ is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion minimal solution $S$). As a specific example, we can solve the $\ell$-subgraph contractibility problem in which the edges of a set $S$ are required to form disjoint connected subgraphs of size at most $\ell$. This problem can be solved in time $O(n^2 f'(k,\ell))$ using the general algorithm. We also show that for $\ell \ge 2$ the problem is NP-complete.