Zero Forcing Number, Maximum Nullity and Path Cover Number for Edge Subdivision Graphs

For a simple graph G the zero forcing number Z(G)  is the minimum number of black vertices initially needed to force all vertices in G black according to the color change rule. The color change rule states that for G with all vertices colored either black or white, if a vertex v is black and an adjacent white vertex w is the only white neighbor of v, then v can force w to be colored black. The maximum nullity M(G) of a graph G is defined to be the largest possible nullity over all symmetric real matrices described by G. It is known that M(G) ≤ Z(G) for all G. The path cover number P(G) of a graph G is the minimum number of induced paths needed to cover all the vertices of G.  To subdivide the edge e between vertices u and v, delete e and add a new vertex w adjacent to exactly u and v.   An edge subdivision graph of G is obtained from G by subdividing zero or more edges of G. We present results on Z(H), M(H), and P(H) for an edge subdivision graph H of G and an open question in the literature is answered.