Spanning Tree Problems on Graph Classes

• Main Authors: Ching-Chi Lin, 林清池
• Other Authors: Gen-Huey Chen
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/12333125022467583386

博士 === 國立臺灣大學 === 資訊工程學研究所 === 95 === A spanning subgraph of a graph G is a subgraph containing all vertices of G. A spanning tree of G is a spanning subgraph of G that is a tree. Spanning trees are not only fundamental in algorithmic graph theory but also practically important in many applications such as communication networks, messages encoding and routing algorithm. Based on different criteria on spanning trees, there are different types of spanning trees. The most famous spanning tree problems are the minimum cost spanning tree problem and the shortest path tree problem. Both problems are solvable in polynomial time. On the other hand, some spanning tree problems are not easy to be solved. For example, the Steiner minimal tree problem and the maximum leaf spanning tree problem are NP-complete. In this dissertation, we study the problems of finding locally connected spanning trees and degree-preserving spanning trees. Both problems have been proven to be NP-complete on general graphs. We design polynomial-time algorithms for these two problems on some graph classes such as directed path graphs, strongly chordal graphs and circular-arc graphs. A locally connected spanning tree of a graph G is a spanning tree T such that the set of all neighbors of v in T induces a connected subgraph of G for every vertex v of G. Given a strong elimination order of a strongly chordal graph, we propose an O(m+n)-time algorithm for determining whether the graph contains a locally connected spanning tree and producing it if it exists. Further, given an intersection model of a circular-arc graph, an O(n)-time algorithm for finding locally connected spanning trees on circular-arc graphs are proposed. A vertex v of G is a degree-preserving vertex if its degree in T is the same as in G. The degree-preserving spanning tree problem is to find a spanning tree T of a connected graph G such that the number of degree-preserving vertices is maximum. Given a strong elimination order of a strongly chordal graph, we show an O(m α(m, n))-time algorithm on strongly chordal graphs, where α is the inverse of Ackermann''s function. Moreover, given a tail order of a directed path graph, an O(m+n)-time algorithm for the degree-preserving spanning tree problem on directed path graphs is proposed. Further, given an intersection model of a circular-arc graph, we show an algorithm on circular-arc graphs that can be completed in O(n^2) time.