An optimal algorithm for the k-fixed-endpoint path cover on proper interval graphs

Mertzios, G.B.; Unger, W.

Abstract

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.

Citation

Mertzios, G., & Unger, W. (2010). An optimal algorithm for the k-fixed-endpoint path cover on proper interval graphs. Mathematics in Computer Science, 3(1), 85-96. https://doi.org/10.1007/s11786-009-0004-y

• Journal Article Type: Article
• Publication Date: Mar 1, 2010
• Deposit Date: Dec 8, 2011
• Publicly Available Date: Jan 10, 2012
• Journal: Mathematics in Computer Science
• Print ISSN: 1661-8270
• Electronic ISSN: 1661-8289
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 3
• Issue: 1
• Pages: 85-96
• DOI: https://doi.org/10.1007/s11786-009-0004-y
• Keywords: Proper interval graph, Perfect graph, Path cover, SNIR matrix, Linear-time algorithm.
• Public URL: https://durham-repository.worktribe.com/output/1502549