RICE UNIVERSITY Graph Coloring, Zero Forcing, and Related Problems by Boris Brimkov A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Doctor of Philosophy APPROVED, THESIS COMMITTEE: Professor of Computational and Applied Mathematics Andrew J. Schaefer Noah Harding Professor of Computational and Applied Mathematics Assistant Professor of Computational and Applied Mathematics C~tneAssociate Professor of Computer Science Houston, Texas May, 2017 ABSTRACT Graph Coloring, Zero Forcing, and Related Problems by Boris Brimkov This thesis investigates several problems related to classical and dynamic color- ing of graphs, and enumeration of graph attributes. In the first part of the thesis, I present new efficient methods to compute the chromatic and flow polynomials of specific families of graphs. The chromatic and flow polynomials count the number of ways to color and assign flow to the graph, and encode information relevant to various physical applications. The second part of the thesis focuses on zero forcing | a dynamic graph coloring process whereby at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. Zero forcing has applications in linear algebra, quantum control, and power network monitoring. A connected forcing set is a connected set of initially colored vertices which forces the entire graph to become colored; the connected forcing number is the cardinality of the smallest connected forcing set. I present a variety of structural results about connected forcing, such as the effects of vertex and edge operations on the connected forcing number, the relations between the connected forcing number and other graph parameters, and the computational complexity of connected forcing. I also give ef- ficient algorithms for computing the connected forcing numbers of different families of graphs, and characterize the graphs with extremal connected forcing numbers. Fi- nally, I investigate several enumeration problems associated with zero forcing, such iii as the exponential growth of certain families of forcing sets, relations of families of forcing sets to matroids and greedoids, and polynomials which count the number of distinct forcing sets of a given size. Acknowledgments I would like to thank my parents for their love and encouragement; my advisor Dr. Illya Hicks for his wisdom, leadership, and enthusiasm which has inspired me to fulfill my research goals; the members of my committee Dr. Andrew Schaefer, Dr. Matthew Knepley, and Dr. Christopher Jermaine for their helpful remarks and for enhancing my background in computational and applied mathematics. I am grate- ful to all my colleagues in the CAAM Department for supporting me throughout the course of my graduate career and providing an enriching work environment. Financial support for this research was provided by the National Science Founda- tion under Grant No. 1450681. Contents Abstract ii List of Illustrations viii 1 Background and literature review 1 1.1 Graph polynomials . 1 1.1.1 The chromatic polynomial . 3 1.1.2 The flow polynomial . 15 1.1.3 Other graph polynomials . 28 1.2 Zero forcing and related problems . 30 1.2.1 Zero forcing . 31 1.2.2 Minimum rank . 33 1.2.3 Power domination . 36 1.2.4 Graph searching, quantum control, and zero forcing variants . 38 1.2.5 Connected zero forcing . 41 2 Preliminaries 45 2.1 Graph definitions . 45 2.2 Graph parameters and operations . 47 2.3 Special types of graphs . 49 3 Characterizations of chromatic and flow polynomials 51 3.1 Characterizations of chromatic polynomials . 52 3.1.1 Generalized vertex join trees . 53 3.1.2 Generalized vertex join cycles . 61 vi 3.1.3 Generalized vertex join cliques . 64 3.2 Characterizations of flow polynomials . 65 3.2.1 Outerplanar graphs . 65 3.2.2 Generalized vertex join cycles . 68 3.3 Computational results . 69 3.3.1 Generalized vertex join trees . 70 3.3.2 Generalized vertex join cycles . 72 3.3.3 Generalized vertex join cliques . 72 3.3.4 Discussion . 74 4 Structural results and complexity of connected forcing 76 4.1 Basic properties and relation to other parameters . 76 4.2 Vertex and edge operations . 79 4.3 Bounds and mandatory vertices . 85 4.4 Computational complexity . 91 5 Characterizations of connected forcing numbers 94 5.1 Graphs with symmetry . 94 5.2 Trees and uniclique graphs . 100 5.3 Unicyclic graphs . 104 5.4 Cactus and block graphs with no pendant paths . 117 6 Graphs with extremal connected forcing numbers 121 6.1 Graphs with Zc(G) = 1 and Zc(G) = n − 1 . 122 6.2 Graphs with Zc(G)=2.......................... 123 6.3 Graphs with Zc(G) = n − 2 . 129 7 Enumeration problems related to zero forcing 152 7.1 Exponential quantities associated with zero forcing . 152 vii 7.2 Connected forcing and matroids . 156 7.3 Zero forcing polynomial . 162 8 Conclusion 177 Bibliography 181 Illustrations 1.1 A four-colored map . 2 1.2 3-colorings of a graph . 4 1.3 Chromatic polynomial of house graph . 6 1.4 Appearance of chromatic polynomial . 8 1.5 Nowhere-zero Z3- and Z4-flows of a graph . 19 1.6 Flow polynomial of house graph . 22 1.7 Zero forcing in a graph . 31 2.1 Vertex join of a graph . 49 2.2 Graph and its dual . 50 3.1 Generalized vertex join of a graph . 52 3.2 Forming a generalized vertex join tree . 54 3.3 Removing bridges of a generalized vertex join tree . 55 3.4 Forming levels in a generalized vertex join tree . 56 3.5 Special subgraphs of a generalized vertex join tree . 57 3.6 Generalized vertex join cycle CS ..................... 62 3.7 CS0 , the underlying graph of CS ..................... 62 3.8 Decomposing a generalized vertex join cycle . 63 3.9 The dual of a generalized vertex join cycle . 69 3.10 Computing P (TS) with Algorithm 1 and Mathematica . 71 3.11 Computing P (TS) with Algorithm 1 for larger graphs . 71 ix 3.12 Computing P (CS) with Algorithm 3 and Mathematica . 73 3.13 Computing P (KS) with Algorithm 4 and Mathematica . 74 4.1 Zc(G) and Z(G) can differ arbitrarily . 77 4.2 Minimum connected forcing set containment . 78 4.3 The connected forcing spread can be arbitrarily large . 81 4.4 Contraction can change the connected forcing number arbitrarily . 82 4.5 Subdivision can make the connected forcing number grow arbitrarily . 84 4.6 Gadget for complexity of connected forcing . 92 5.1 Sun graph and minimum connected forcing set . 100 5.2 Uniclique graphs G for which Z(G) = Zc(G) . 104 5.3 A unicyclic graph and a minimum connected forcing set . 115 5.4 Unicyclic graphs G for which Z(G) = Zc(G) . 117 6.1 Graphs G of two parallel paths with Zc(G) = 2, case 1 . 125 6.2 Graphs G of two parallel paths with Zc(G) = 2, case 2 . 126 6.3 Forbidden induced subgraphs for Z(G) ≥ n − 2 . 129 6.4 Graphs G with Zc(G) = n − 2, κ(G) = 1, case 1 . 132 6.5 Graphs G with Zc(G) = n − 2, κ(G) = 1, case 2 . 134 6.6 Graphs G with Zc(G) = n − 2, κ(G) = 1, case 3 . 136 6.7 Graphs G with Zc(G) = n − 2, κ(G) = 1, case 4 . 139 6.8 Graphs G with Zc(G) = n − 2, κ(G) ≥ 2, case 1 . 141 6.9 Graphs G with Zc(G) = n − 2, κ(G) ≥ 2, case 2 . 149 7.1 A tree for which zc(T ; 8) is computed . 172 7.2 All connected forcing sets of size 8 of a tree . 173 7.3 A graph with a unique minimum connected forcing set . 174 1 Chapter 1 Background and literature review This chapter surveys the literature and historical development of chromatic and flow polynomials, zero forcing, and related topics. See Chapter 2 for graph theoretic notions and operations used throughout this discussion. 1.1 Graph polynomials Algebraic graph theory studies properties of graphs by algebraic means. This ap- proach often leads to elegant proofs and at times reveals deep and unexpected con- nections between graph theory and algebra. In the last few decades, algebraic graph theory has developed rapidly, generating a substantial body of literature; the mono- graphs of Biggs [1] and Godsil and Royle [2] are standard sources on the subject. A central topic of algebraic graph theory is the study of polynomials associated with graphs. These polynomials contain important information about the structure and properties of graphs, and enable its extraction by algebraic methods. In partic- ular, the values of graph polynomials at specific points, as well as their coefficients, roots, and derivatives, often have meaningful interpretations. The study of graph polynomials was motivated by the Four Color Conjecture, which states that any map can be \face-colored" using four colors so that neighbor- ing regions do not share the same color; see Figure 1.1 for an example. In 1912, Birkhoff [4] introduced a polynomial which counts the ways to face-color a planar 2 Figure 1.1 : A map colored with four colors (adapted from [3]) graph and attempted to prove the Four Color Conjecture by analyzing the roots of this polynomial. For planar graphs, the concept of face-coloring is equivalent to \vertex-coloring", and in 1932, Whitney [5] generalized Birkhoff's polynomial to count vertex-colorings of general graphs; this polynomial is known today as the chromatic polynomial. The Four Color Conjecture was proved true in 1976 by Appel and Haken [6] with the help of a computer, though an analytic proof in the vein of Birkhoff's attempt is still being sought.