Zero Forcing Sets and Bipartite Circulants

In this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices and we describe some of their properties. Notably, we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix. The main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound.Comment: 22 pages, 13 figure

A computational comparison of compact MILP formulations for the zero forcing number

Consider a graph where some of its vertices are colored. A colored vertex with a single uncolored neighbor forces that neighbor to become colored. A zero forcing set is a set of colored vertices that forces all vertices to become colored. The zero forcing number is the size of a minimum forcing set. Finding the minimum forcing set of a graph is NP-hard. We give a new compact mixed integer linear programming formulation (MILP) for this problem, and analyse this formulation and establish relation to an existing compact formulation and to two variants. In order to solve large size instances we propose a sequential search algorithm which can also be used as a heuristic to derive upper bounds for the zero forcing number. A computational study using Xpress (a MILP solver) is conducted to test the performances of the discussed compact formulations and the sequential search algorithm. We report results on cubic, Watts-Strogatz and randomly generated graphs with 10, 20 and 30 vertices.publishe

Problems in graph theory and partially ordered sets

This dissertation answers problems in three areas of combinatorics - processes on graphs, graph coloring, and antichains in a partially ordered set.First we consider Zero Forcing on graphs, an iterative infection process introduced by AIM Minimum Rank - Special Graphs Workgroup in 2008. The Zero Forcing process is a graph infection process obeying the following rules: a white vertex is turned black if it is the only white neighbor of some black vertex. The Zero Forcing Number of a graph is the minimum cardinality over all sets of black vertices such that, after a finite number of iterations, every vertex is black. We establish some results about the zero forcing number of certain graphs and provide a counter example of a conjecture of Gentner and Rautenbach. This chapter is joint with Gabor Meszaros, Antonio Girao, and Chapter 3 appears in Discrete Math, Vol. 341(4).In the second part, we consider problems in the area of Dynamic Coloring of graphs. Originally introduced by Montgomery in 2001, the r-dynamic chromatic number of a graph G is the least k such that V(G) is properly colored, and each vertex is adjacent to at least r different colors. In this coloring regime, we prove some bounds for graphs with lattice like structures, hypercubes, generalized intervals, and other graphs of interest. Next, we establish some of the first results in the area of r-dynamic coloring on random graphs. The work in this section is joint with Peter van Hintum.In the third part, we consider a question about the structure of the partially ordered set of all connected graphs. Let G be the set of all connected graphs on vertex set [n]. Define the partial ordering \u3c on G as follows: for G,H G let G \u3c H if E(G) E(H). The poset (G

Novel Techniques for the Zero-Forcing and p-Median Graph Location Problems

This thesis presents new methods for solving two graph location problems, the p-Median problem and the zero-forcing problem. For the p-median problem, I present a branch decomposition based method that finds the best p-median solution that is limited to some input support graph. The algorithm can be used to either find an integral solution from a fractional linear programming solution, or it can be used to improve on the solutions given by a pool of heuristics. In either use, the algorithm compares favorably in running time or solution quality to state-of-the-art heuristics. For the zero-forcing problem, this thesis gives both theoretical and computational results. In the theoretical section, I show that the branchwidth of a graph is a lower bound on its zero-forcing number, and I introduce new bounds on the zero-forcing iteration index for cubic graphs. This thesis also introduces a special type of graph structure, a zero-forcing fort, that provides a powerful tool for the analysis and modeling of zero-forcing problems. In the computational section, I introduce multiple integer programming models for finding minimum zero-forcing sets and integer programming and combinatorial branch and bound methods for finding minimum connected zero-forcing sets. While the integer programming methods do not perform better than the best combinatorial method for the basic zero-forcing problem, they are easily adapted to the connected zero-forcing problem, and they are the best methods for the connected zero-forcing problem