The Subgraph Homeomorphism Problem

Abstract

The problem investigated in this thesis is that of finding homeomorphic images of a given graph, called the pattern graph, in a larger graph. A homeomorphism is a pair of mappings, (v,a), suc that v maps the nodes of the pattern graph to nodes of the larger graph, and a maps the edges of the mattern graph to (edge or node) disjoint paths in the larger graph. A homeomorphism represents a similarity of structure between the graphs involved. Therefore, it is an important concept for both graph theory and applications such as programming schema. We give a formal definition of the subgraph homeomorphism problem. In our investigation, we focus on algorithsm which depend on the pattern graph and allow the node mapping, v, to be partially or totally specified. Reductions between node disjoint and edge disjoint formulations of the problem are discussed. Also, reductions faciliating the solution of given subgraph homeomorphism problems are formulated. A linera time algorithm for finding a cycle in a graph containing three given nodes of the graph is presented. FInally, the two disjoint paths problem, an open problem, is discussed in detail.