# Destroying Bicolored $P_3$s by Deleting Few Edges

## TL;DR

This paper investigates the computational complexity of removing edges to eliminate bicolored $P_3$ subgraphs in a graph with red and blue edges, providing hardness results, polynomial cases, and efficient algorithms.

## Contribution

It introduces the Bicolored $P_3$ Deletion problem, proves its NP-hardness, explores special cases with polynomial solutions, and offers fixed-parameter algorithms and kernelization results.

## Key findings

- NP-hardness of Bicolored $P_3$ Deletion
- Polynomial-time solvability for graphs without bicolored triangles
- Fixed-parameter algorithm with $O(1.84^k)$ time complexity

## Abstract

We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$.