The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs. History: This “Challenge” paper was invited by the Editor in Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida. Funding: This work was supported by the National Science Foundation [Grant DMS-1720225]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1219 .

Zero forcing is a graph coloring process based on the following color change rule: all vertices of a graph [Formula: see text] are initially colored either blue or white; in each timestep, a white vertex turns blue if it is the only white neighbor of some blue vertex. A zero forcing set of [Formula: see text] is a set of blue vertices such that all vertices eventually become blue after iteratively applying the color change rule. The zero forcing number [Formula: see text] is the cardinality of a minimum zero forcing set. In this paper, we propose novel exact algorithms for computing [Formula: see text] based on formulating the zero forcing problem as a two-stage Boolean satisfiability problem. We also propose several heuristics for zero forcing based on iteratively adding blue vertices which color a large part of the remaining white vertices. These heuristics are used to speed up the exact algorithms and can also be of independent interest in approximating [Formula: see text]. Computational results on various types of graphs show that, in many cases, our algorithms offer a significant improvement on the state-of-the-art algorithms for zero forcing. Summary of Contribution: This paper proposes novel algorithms and heuristics for an NP-hard graph coloring problem that has numerous applications. Our exact methods combine Boolean satisfiability modeling with a constraint generation framework commonly used in operations research. The paper also includes an analysis of the facets of the polytope associated with this problem and decomposition techniques which can reduce the size of the problem. Our computational approaches are implemented and tested on a wide variety of graphs and are compared with the state-of-the-art algorithms from the literature. We show that our proposed algorithms based on Boolean satisfiability, in conjunction with the heuristics and order-reduction techniques, yield a significant speedup in some cases.

Abstract The concept of branch decomposition was first introduced by Robertson and Seymour in their proof of the Graph Minors Theorem, and can be seen as a measure of the global connectivity of a graph. Since then, branch decomposition and branchwidth have been used for computationally solving combinatorial optimization problems modeled on graphs and matroids. General branchwidth is the extension of branchwidth to any symmetric submodular function defined over a finite set. General branchwidth encompasses graphic branchwidth, matroidal branchwidth, and rankwidth. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. It was shown in [I. V. Hicks. Graphs, branchwidth, and tangles! Oh my! Networks, 45:55‐60, 2005] that a tangle basis of order k is coextensive to a tangle of order k. In this paper, we revisit the construction of tangle bases computationally for other branchwidth parameters and show that the tangle basis approach is still competitive for computing optimal branch decompositions for general branchwidth.

The status of a vertex v in a connected graph is the sum of the distances from v to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate the status sequences of trees. Particularly, we show that it is NP-complete to decide whether there exists a tree that has a given sequence of integers as its status sequence. We also present some new results about trees whose status sequences are comprised of a few distinct numbers or many distinct numbers. In this direction, we show that any status injective tree is unique among trees. Finally, we investigate how orbit partitions and equitable partitions relate to the status sequence.