On the complexity of failed zero forcing
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 socalled 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 NPhard to compute. Our proof also works for a related graph invariant which is called the skew failed forcing number.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.00211
 Bibcode:
 2016arXiv160900211S
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 5 pages