A Linear Time Algorithm for Finding a Minimum Spanning Tree with Non-Terminal Set VNT on Series-Parallel Graphs

Shin-ichi NAKAYAMA  Shigeru MASUYAMA  

IEICE TRANSACTIONS on Information and Systems   Vol.E102-D   No.4   pp.826-835
Publication Date: 2019/04/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2018EDP7232
Type of Manuscript: PAPER
Category: Fundamentals of Information Systems
spanning tree,  series-parallel graph,  algorithm,  

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Given a graph G=(V,E), where V and E are vertex and edge sets of G, and a subset VNT of vertices called a non-terminal set, the minimum spanning tree with a non-terminal set VNT, denoted by MSTNT, is a connected and acyclic spanning subgraph of G that contains all vertices of V with the minimum weight where each vertex in a non-terminal set is not a leaf. On general graphs, the problem of finding an MSTNT of G is NP-hard. We show that if G is a series-parallel graph then finding an MSTNT of G is linearly solvable with respect to the number of vertices.