Dominating sequences in grid-like and toroidal graphs

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.Comment: 17 pages 3 figure

Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k

We consider the family of graphs whose vertex set is Z^k where two vertices are connected by an edge when their l\infty-distance is 1. We prove the optimal vertex isoperimetric inequality for this family of graphs. That is, given a positive integer n, we find a set A \subset Z^k of size n such that the number of vertices who share an edge with some vertex in A is minimized. These sets of minimal boundary are nested, and the proof uses the technique of compression. We also show a method of calculating the vertex boundary for certain subsets in this family of graphs. This calculation and the isoperimetric inequality allow us to indirectly find the sets which minimize the function calculating the boundary.Comment: 19 pages, 2 figure

Treewidth and related graph parameters

For modeling some practical problems, graphs play very important roles. Since many modeled problems can be NP-hard in general, some restrictions for inputs are required. Bounding a graph parameter of the inputs is one of the successful approaches. We study this approach in this thesis. More precisely, we study two graph parameters, spanning tree congestion and security number, that are related to treewidth. Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G connecting two components of T − e. The edge congestion of G in T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion of G in its spanning trees. In this thesis, we show the spanning tree congestion for the complete k-partite graphs, the two-dimensional tori, and the twodimensional Hamming graphs. We also address lower bounds of spanning tree congestion for the multi-dimensional hypercubes, the multi-dimensional grids, and the multi-dimensional Hamming graphs. The security number of a graph is the cardinality of a smallest vertex subset of the graph such that any “attack” on the subset is “defendable.” In this thesis, we determine the security number of two-dimensional cylinders and tori. This result settles a conjecture of Brigham, Dutton and Hedetniemi [Discrete Appl. Math. 155 (2007) 1708–1714]. We also show that every outerplanar graph has security number at most three. Additionally, we present lower and upper bounds for some classes of graphs.学位記番号：工博甲39

Decision problems for 3-manifolds and their fundamental groups

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.Comment: 31 pages, final versio

Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).Comment: 42 pages, 8 figures. Version 2: 47 pages, 8 figures; minor revisions and reformatting; to appear in Geom. Topo