Loop zero forcing on graphs

The loop zero forcing number of a graph is based on the loop color change rule: If a white vertex v is the only white neighbor of a black vertex u, then change the color of v to black. If the neighbors of a white vertex v are all black, then change the color of v to black. A minimum loop zero forcing set is a set of black vertices that changes the entire graph to black and has a minimum cardinality. The loop zero forcing number of a graph is the cardinality of a minimum loop zero forcing set of the graph. The loop propagation time is the least number of rounds; it takes for a minimum loop zero forcing set to change the whole graph to black. A failed loop zero forcing number is the largest set that fails to change the entire graph to black. The loop throttling number minimizes the sum of a loop zero forcing set and its loop propagation time. In this work, we establish various properties of the loop zero forcing number, loop propagation time, loop throttling number, and failed loop zero forcing number of graphs. Graphs giving the extreme values of these concepts are also characterized.