On the complexity of failed zero forcing

TL;DR

This paper investigates the computational complexity of the failed forcing number in graphs, proving it is NP-hard to compute, which advances understanding of graph forcing processes and related invariants.

Contribution

It establishes the NP-hardness of computing the failed forcing number and extends the result to the skew failed forcing number, answering a recent open question.

Findings

Failed forcing number is NP-hard to compute.

The NP-hardness result applies to the skew failed forcing number.

The proof addresses an open problem in graph forcing theory.

Abstract

Let $G$ be a simple graph whose vertices are partitioned into two subsets, called filled vertices and empty vertices. A vertex $v$ is said to be forced by a filled vertex $u$ if $v$ is a unique empty neighbor of $u$. If we can fill all the vertices of $G$ by repeatedly filling the forced ones, then we call an initial set of filled vertices a forcing set. We discuss the so-called failed forcing number of a graph, which is the largest cardinality of a set which is not forcing. Answering the recent question of Ansill, Jacob, Penzellna, Saavedra, we prove that this quantity is NP-hard to compute. Our proof also works for a related graph invariant which is called the skew failed forcing number.