Relationships between minimum rank problem parameters for cobipartite graphs
Louis Deaett, Derek Young
TL;DR
The paper investigates the relationship between the minimum rank problems for graphs and for zero-nonzero patterns, focusing on cobipartite graphs. It shows that over infinite fields a tight link exists between the triangle number of a pattern and the graph's zero forcing number, mirroring the graph-pattern minimum rank relationship and tying this to maximum nullity via $M(G)$. A key result expresses the enhanced zero forcing number as $\hat{Z}(G) = |G| - \min\{ \mathrm{tri}(\mathcal{A}) : \mathcal{A} \in \mathcal{S}_{\mathrm{znz}}(G) \}$, and the work analyzes how saturation and loopings govern when the parameters align, with consequences for cobipartite graphs and related zero forcing variants. The findings include a $15$-vertex counterexample showing a universal gap between maximum nullity and zero forcing number and a discussion of how unsaturated vertices affect these quantities.
Abstract
For a simple graph, the minimum rank problem is to determine the smallest rank among the symmetric matrices whose off-diagonal nonzero entries occur in positions corresponding to the edges of the graph. Bounds on this minimum rank (and on an equivalent value, the maximum nullity) are given by various graph parameters, most notably the zero forcing number and its variants. For a matrix, replacing each nonzero entry with the symbol \(\ast\) gives its zero-nonzero pattern. The associated minimum rank problem is to determine, given only this pattern, the smallest possible rank of the matrix. The most fundamental lower bound on this minimum rank is the triangle number of the pattern. A cobipartite graph is the complement of a bipartite graph; its vertices can be partitioned into two cliques. Such a graph corresponds to a zero-nonzero pattern in a natural way. Over an infinite field, the minimum rank of the graph and that of the pattern obey a simple relationship. We show that this same relationship is followed by the zero forcing number of the graph and the triangle number of the pattern. This has implications for the relationship between the two minimum rank problems. We also explore how, for cobipartite graphs, variants of the zero forcing number and other parameters important to the minimum rank problem are related, as well as how, for graphs in general, these parameters can be interpreted in terms of the zero-nonzero patterns of the symmetric matrices associated with the graph.