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Full-Text Articles in Physical Sciences and Mathematics
Bounds For The Zero Forcing Number Of Graphs With Large Girth, Randy Davila, Franklin Kenter
Bounds For The Zero Forcing Number Of Graphs With Large Girth, Randy Davila, Franklin Kenter
Theory and Applications of Graphs
The zero-forcing number, Ζ(G) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Ζ(G) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ Ζ(G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Ζ(G) as a …
Dynamic Approach To K-Forcing, Yair Caro, Ryan Pepper
Dynamic Approach To K-Forcing, Yair Caro, Ryan Pepper
Theory and Applications of Graphs
The k-forcing number of a graph is a generalization of the zero forcing number. In this note, we give a greedy algorithm to approximate the k-forcing number of a graph. Using this dynamic approach, we give corollaries which improve upon two theorems from a recent paper of Amos, Caro, Davila and Pepper [2], while also answering an open problem posed by Meyer [9].