A Polynomial Kernel for Proper Helly Circular-arc Vertex Deletion
Akanksha Agrawal, Satyabrata Jana, Abhishek Sahu
TL;DR
This work resolves the PHCAVD problem by producing a polynomial kernel, advancing the kernelization frontier for vertex deletion problems targeting graph classes defined by forbidden obstructions. The authors combine an efficient modulator approach with a rich obstruction framework: run a $6$-approximation to obtain a compact modulator $T_1$, extend to a $5$-redundant solution $M$ and form a nice modulator $T=T_1\cup M$, ensuring every obstruction of interest intersects $T$ in a controlled way. They then exploit a nice clique partition of the residual graph $G-T$ and a carefully designed marking scheme (Mark-1) to bound the size of each clique and the number of components; a sequence of reduction rules (e.g., bounding cliques, handling claws, and min-cuts between cliques) yields a kernel of size $O(k^{88})$. The polynomial kernel demonstrates the practical viability of preprocessing PHCAVD instances and provides a foundation for further refinements in obstruction-based kernelization for circular-arc graph modification problems, with implications for related graph classes and Deletion problems.
Abstract
A proper Helly circular-arc graph is an intersection graph of a set of arcs on a circle such that none of the arcs properly contains any other arc and every set of pairwise intersecting arcs has a common intersection. The Proper Helly Circular-arc Vertex Deletion problem takes as input a graph $G$ and an integer $k$, and the goal is to check if we can remove at most $k$ vertices from the graph to obtain a proper Helly circular-arc graph; the parameter is $k$. Recently, Cao et al.~[MFCS 2023] obtained an FPT algorithm for this (and related) problem. In this work, we obtain a polynomial kernel for the problem.