On the (Positive) Zero Forcing Numbers of the Graphs

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph. We show correspondingly that the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. Using this fact we show via a purely graph theoretical proof that the positive zero forcing number of any outerplanar graph equals the treecover number of the graph. Also motivating by one the most interesting open questions that has been proposed by de Verdi ere in 1998, we stablish a connection between the positive zero forcing number and the chromatic number of a graph.

Keywords:

Zero forcing number , positive zero forcing number , path cover number , tree cover number