| Summary: | 博士 === 國立中央大學 === 資訊工程研究所 === 89 === In this dissertation, several geodetic problems in graphs are considered. The motivation of studying geodetic connectivity and hinge vertices in a graph arises naturally from network design and analysis. We first look at the algorithmic complexities of computing geodetic connectivity and finding all hinge vertices in a graph. We present an O(nm) time algorithm for solving this problem in an arbitrary graph. In particular, we are interested to know if there exist special classes of graphs with special properties such that the time complexity of determining whether a graph has geodetic connectivity k can be reduced. We investigate this topic on some special classes of chordal graphs. Linear time algorithms are developed for strongly chordal graphs, ptolemaic graphs, and interval graphs, and an O(n^2) time algorithm is proposed for undirected path graphs. Because many interconnection networks can be constructed using line graph iterations, we also study how to recognize line graphs without hinge vertices. Moreover, we provide several operations of graphs that allow a graph to expand in scale without destroying geodetically connectivity.
Next, we deal with the topic on constructing a specific vertex ordering of an AT-free graph. We show that every connected AT-free graph admits a vertex ordering that we call a 2-cocomparability ordering. The ordering presented here can be defined by using the term of distance metric, and can be exploited for algorithmic purposes. Two techniques called involutive sequence and 2-sweep LexBFS are introduced for constructing such an ordering for AT-free graphs. This ordering generalizes the cocomparability ordering achievable for cocomparability graphs. According to this ordering, we show that every power of an AT-free graph is indeed a cocomparability graph. Also, it leads to that the distance k-domination, k-stability, and the graph k-clustering problems for k>=2 on AT-free graphs can be solved in a more efficient way. In particular, the vertex ordering produced from the 2-sweep LexBFS can be used for constructing a tree 4-spanner of an AT-free graph and for recognizing a class of special AT-free graphs called bipartite permutation graphs in linear time.
Besides, we study the characterization of chordal bipartite graphs and show that the all-pairs-shortest-length problem on chordal bipartite graphs can be solved in O(n^2) optimal time. We further investigate the algorithmic complexity of a graph clustering problem that determine whether there is a partition of a graph into certain number clusters of vertices such that the diameter of subgraph induced by each cluster does not exceed a prespecified bound. Some NP-complete and polynomial results are derived for several classes of special graphs.
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