The kk-visibility Localization Game

We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number $\zeta_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded $\zeta_k$ values. Extending results known for the localization number, we show that for $k\geq 2$, every tree contains a subdivision with $\zeta_k = 1$. For many $n$, we give the exact value of $\zeta_k$ for the $n \times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that $\zeta_i \neq \zeta_j$ for all distinct choices of $i$ and $j.

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