Algorithms for Graphs of (Locally) Bounded Treewidth

Many real-life problems can be modeled by graph-theoretic problems. These graph problems are usually NP-hard and hence there is no efficient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of graph for which several NP-complete problems can be solved in polynomial time. Graphs of bounded treewidth, which generalize trees, show good algorithmic properties similar to those of trees. Using ideas developed for tree algorithms, Arnborg and Proskurowski introduced a general dynamic programming approach which solves many problems such as dominating set, vertex cover and independent set. Others used this approach to solve other NP-hard problems. Matoušek and Thomas applied this approach to solve the subgraph isomorphism problem when the source graph has bounded degree and the host graph has bounded treewidth. In this thesis, we introduce a new property for graphs called log-bounded fragmentation, by which we mean after removing any set of at most k vertices the number of connected components is at most O(k logn), where n is the number of vertices of the graph. We then extend the result of Matoušek and Thomas to the case in which the source graph is a log-bounded fragmentation graph and the host graph has bounded treewidth. Besides this result, we demonstrate how bounded fragmentation might be used to measure the reliability of a network. As the class of graphs of bounded treewidth is of limited size, we need to solve NP-hard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has locally bounded treewidth if for each vertex v of G, the treewidth of the subgraph of G induced on all vertices of distance at most r from v is only a function of r, called local treewidth. So far the only graphs determined to have small local treewidth are planar graphs. In this thesis, we prove that the local treewidth of K3,3-minor-free or K5-minor-free graphs is also bounded above by 3r+4. Using this result, we extend several polynomial-time approximation algorithms on planar graphs to these graphs. Algorithms on graphs of bounded treewidth also can be extended to graphs of locally bounded treewidth. As an example, we demonstrate how the subgraph isomorphism problem on graphs of locally bounded treewidth can be solved in polynomial time, when the source graph is a log-bounded fragmentation graph and has constant diameter.