Zero forcing number, maximum nullity, and path cover number of subdivided graphs

Abstract

Abstract. The zero forcing number, maximum nullity and path cover number of a (simple, undirected) graph are parameters that are important in the study of minimum rank problems. We investigate the effects on these graph parameters when an edge is subdivided to obtain a so-called edge subdivision graph. An open question raised by Barrett et al. is answered in the negative, and we provide additional evidence for an affirmative answer to another open question in that paper [W. Barrett, R. Bowcutt, M. Cutler, S. Gibelyou, and K. Owens. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530-563, 2009.]. It is shown that there is an independent relationship between the change in maximum nullity and zero forcing number caused by subdividing an edge once. Bounds on the effect of a single edge subdivision on the path cover number are presented, conditions under which the path cover number is preserved are given, and it is shown that the path cover number and the zero forcing number of a complete subdivision graph need not be equal.

The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

“…, then change the signs of edges (1, 2 ), (2, 3 ) and (3,4 ) to zero according to Rule C(c). We obtain that…”

“…Clearly, the solution of the minimum rank problem is equivalent to determining the maximum nullity problem. In order to study the minimum rank problem of a graph, some graph parameters have been used to bound the maximum nullity of a graph, and the relationship between the maximum nullity and the related graph parameters has received considerable attention (see [1][2][3][4][5][6][7][8][9]). The zero forcing number provides a method to bound the maximum nullity for a sign pattern.…”

The maximum nullity of a complete subdivision graph is equal to its zero forcing number

“…The equality M(F,G) = Z(G) was established for graphs that have a Hamilton path in [5] (with an easy formula for the value) and for graphs that do not have a bridge in [10] (without a formula for the value). Our main result, Theorem 1.3, extends this identity to all graphs G and gives an easily computed formula for Z(G).…”

“…Let K be the family of bipartite graphs G = (V (G), E(G)) such that there is a bipartition of the vertices V (G) = X∪ Y with deg x ≤ 2 for all x ∈ X [10]. Clearly every complete edge subdivision graph is in K. A graph G ∈ K is special if for every field F there exists a matrix A ∈ S(F, G) such that:…”

“…We will use results from [5] and [10]. Since the proof of the next result uses a key idea and is very brief, it is included.…”