Modification Problems toward Proper (Helly) Circular-arc Graphs

TL;DR

This paper introduces a fixed-parameter tractable algorithm for the proper circular-arc vertex deletion problem, advancing the understanding of modification problems toward proper Helly circular-arc graphs.

Contribution

It provides a novel $9^k imes n^{O(1)}$-time algorithm for a key graph modification problem, resolving an open question and extending to proper Helly circular-arc graphs.

Findings

Developed a $9^k imes n^{O(1)}$-time algorithm for vertex deletion.

Resolved an open problem in the field.

Extended structural insights to proper Helly circular-arc graphs.

Abstract

We present a $9^k\cdot n^{O(1)}$-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [arXiv:2001.06867]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs.