Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

TL;DR

This paper presents a subexponential-time algorithm for testing C-Planarity in embedded flat clustered graphs with bounded face size and introduces an FPT algorithm based on embedded-width and cluster connectivity.

Contribution

It provides the first subexponential algorithm for C-Planarity in certain embedded graphs and establishes fixed-parameter tractability using embedded-width and cluster parameters.

Findings

Subexponential-time algorithm for C-Planarity with bounded face size.

C-Planarity is fixed-parameter tractable with embedded-width and cluster count.

New approach using embedded tree decomposition for clustered graph planarity.

Abstract

The C-Planarity problem asks for a drawing of a $\textit{clustered graph}$, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for $\textit{embedded flat clustered graphs}$, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of $\textit{embedded tree decomposition}$ in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.