The Parameterized Complexity of Problems on Outer k-Planar Graphs

2026-07-29

AI Summary

This study investigates the parameterized complexity of classical graph problems on outer k-planar graphs, parameterized by k, under the assumption that an outer k-planar embedding of the input graph is given. By integrating techniques from parameterized complexity theory, mim-width analysis, and structural graph properties, the authors establish that the mim-width of outer k-planar graphs is at most k + 2. They further demonstrate tight connections between outer k-planarity and other graph parameters: graphs of cutwidth at most k are outer 2k-planar, and those with feedback edge set number at most k are outer 6k-planar. Leveraging these structural insights, the paper shows that many problems—including List Coloring—are fixed-parameter tractable on outer k-planar graphs, while a few—such as Binary CSP and Scattered Set—remain intractable, thereby providing a systematic characterization of how outer k-planar structure influences computational complexity.

Abstract

A graph is outer k-planar if it admits a straight-line drawing in which all vertices lie on a circle and every edge is crossed by at most k other edges. We study the parameterized complexity of a broad collection of graph problems on outer k-planar graphs, with k as the parameter. Many graph problems are known to be XALP-hard when parameterized by treewidth or outerplanarity, and XNLP-hard when parameterized by pathwidth. We show that only a few such problems, including Binary CSP and Scattered Set, remain intractable on outer k-planar graphs, whereas a large class of the others become fixed-parameter tractable in this setting, assuming that an outer k-planar drawing of the input graph is given. These include List Coloring, Capacitated Dominating Set, Capacitated Vertex Cover, Target Outdegree Orientation, and Target Set Selection, among others. In addition to the algorithmic and complexity results, we establish several structural results. We show that outer k-planar graphs have mim-width at most k+2, that graphs of cut-width at most k are outer 2k-planar, and that graphs of feedback edge set number at most k are outer 6k-planar. We also show that many graph parameters are incomparable with outer k-planarity, thereby clarifying its position within the graph parameter hierarchy.

Problem

• outer k-planar graphs
• parameterized complexity
• graph problems
• fixed-parameter tractability
• XALP-hardness

Innovation

• outer k-planar graphs
• parameterized complexity
• fixed-parameter tractability
• mim-width
• graph drawing