Pathwidth-One Vertex Explosion and Co-Path Problems

2026-09-25

AI Summary

This study investigates the Bipartite One-sided Vertex Explosion (BOVE) problem and the restricted Path-width One Vertex Explosion (POVE) problem. Methodologically, it establishes, for the first time, bidirectional parameter-preserving reductions between BOVE and Co-Path Set, as well as between restricted POVE and Co-Path Packing, thereby overcoming the limitations of previously independent solution approaches. Furthermore, the theoretical analysis leverages techniques from parameterized algorithm design and kernelization. Ultimately, this work yields improved parameterized algorithms and a tighter kernel for BOVE, offering a novel paradigm for solving related problems.

Abstract

In the Bipartite One-sided Vertex Explosion problem (BOVE), the input is a bipartite graph $G = (T, B, E)$ and a nonnegative integer $k$. The goal is to decide whether there is a set $S \subseteq B$ of size at most $k$ such that exploding every vertex in $S$ results in a graph with pathwidth at most one. The restricted Pathwidth One Vertex Explosion problem (restricted POVE) is a generalization of BOVE. The input to this problem is a graph $G$, a nonnegative integer $k$, and a set of vertices $X \subseteq V($ The goal is to decide whether there is a set $S \subseteq X$ of size at most $k$ such that exploding every vertex in $S$ results in a graph with pathwidth at most one. In this paper, we show parameter-preserving reductions in both directions between BOVE and Co-Path Set, and between restricted POVE and restricted Co-Path Packing. These reductions yield improved parameterized algorithms for BOVE and restricted POVE and an improved kernel for BOVE.

Problem

• Bipartite One-sided Vertex Explosion
• Pathwidth-One Vertex Explosion
• Co-Path Set
• Parameterized Complexity

Innovation

• Parameter-preserving reduction
• Vertex Explosion
• Co-Path Set
• Parameterized algorithm
• Kernelization