A note on the computational complexity of weak saturation

TL;DR

This paper demonstrates that calculating the weak saturation number for a host graph with respect to a triangle pattern is computationally hard, linking weak saturation to shellability of simplicial complexes.

Contribution

It establishes the computational hardness of weak saturation for triangles and connects it to the shellability of simplicial complexes, providing new insights into the problem's complexity.

Findings

Weak saturation number determination is NP-hard for triangles.

Connection established between weak saturation and shellability.

Provides a new perspective on the complexity of weak saturation problems.

Abstract

We prove that determining the weak saturation number of a host graph $F$ with respect to a pattern graph $H$ is already a computationally hard problem when $H$ is the triangle. As our main tool we establish a connection between weak saturation and shellability of simplicial complexes.