Title: Leaky Forcing: Sensor Allocation In a Fault-Tolerant Setting

Abstract Zero forcing emulates the linear-algebraic properties of backsolving linear equations over any matrix whose sparsity pattern matches a given graph G. Hence, zero forcing has applications in sensor allocation, quantum control and even leader-follower dynamics.

We study a recent variation of zero forcing called leaky forcing.  Zero forcing is a propagation process on a network whereby some nodes are initially blue with all others white. Blue vertices can ``force'' a white neighbor to become blue if all other neighbors are blue. The goal is to find the minimum number of initially blue vertices to eventually force all vertices blue after exhaustively applying the forcing rule above. 

Leaky forcing is a fault-tolerant variation of zero forcing where certain vertices (not necessarily initially blue) cannot force. The goal in this context is to find the minimum number of initially blue vertices needed that can eventually force all vertices to be blue, {\it regardless} of which small number of vertices can't force. This work extends results from zero forcing in terms of leaky forcing. In particular, we provide a complete determination of leaky forcing numbers for all unicyclic graphs and robust upper bounds for generalized Petersen graphs. We also provide bounds for the effect of both edge removal and vertex removal on the $\ell$-leaky forcing number. Finally, we completely characterize connected graphs that have the minimum and maximum possible $1$-leaky forcing number (i.e., when $Z_{(1)}(G) = 2$ and when $\Zone{G} = |V(G)|-1$).

This is joint work with 

Beth Bjorkman, Lei Cao, Franklin Kenter, Ryan Moruzzi, Carolyn Reinhart and Violeta Vasilevska and is part of the AIM Mathematical Research Communities.