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

Related Orderings of AT-Free Graphs

An ordering of a graph G is a bijection of V(G) to {1, . . . , |V(G)|}. In this thesis, we consider the complexity of two types of ordering problems. The first type of problem we consider aims at minimizing objective functions related to an ordering of the graph. We consider the problems Cutwidth, Imbalance, and Optimal Linear Arrangement. We also consider a problem of another type: S-End-Vertex, where S is one of the following search algorithms: breadth-first search (BFS), lexicographic breadth-first search (LBFS), depth-first search (DFS), and maximal neighbourhood search (MNS). This problem asks if a specified vertex can be the last vertex in an ordering generated by S. We show that, for each type of problem, orderings for one problem may be related to orderings for another problem of that type. We show that there is always a cutwidth-minimal ordering where equivalence classes of true twins are grouped for any graph, where true twins are vertices with the same closed neighbourhood. This enables a fixed-parameter tractable (FPT) algorithm for Cutwidth on graphs parameterized by the edge clique cover number of the graph and a new parameter, the restricted twin cover number of the graph. The restricted twin cover number of the graph generalizes the vertex cover number of a graph, and is the smallest value k ≥ 0 such that there is a twin cover of the graph T and k−|T| non-trivial components of G−T. We show that there is also always an imbalance-minimal ordering where equivalence classes of true twins are grouped for any graph. We show a polynomial time algorithm for this problem on superfragile graphs and subsets of proper interval graphs, both subsets of AT-free graphs. An asteroidal triple (AT) is a triple of independent vertices x, y, z such that between every pair of vertices in the triple, there is a path that does not intersect the closed neighbourhood of the third. A graph without an asteroidal triple is said to be AT-free. We also provide closed formulas for Imbalance on some small graph classes. In the FPT setting, we improve algorithms for Imbalance parameterized by the vertex cover number of the input graph and show that the problem does not have a polynomially sized kernel for the same parameter number unless NP ⊆ coNP/poly. We show that Optimal Linear Arrangement also has a polynomial algorithm for superfragile graphs and an FPT algorithm with respect to the restricted twin cover number. Finally, we consider S-End-Vertex, for BFS, LBFS, DFS, and MNS. We perform the first systematic study of the problem on bipartite permutation graphs, a subset of AT-free graphs. We show that for BFS and MNS, the problem has a polynomial time solution. We improve previous results for LBFS, obtaining a linear time algorithm. For DFS, we establish a linear time algorithm. All the results follow from the linear structure of bipartite permutation graphs