A Linear Time Algorithm for Finding a Minimum Spanning Tree with Non-Terminal Set <i>V_NT</i> on Series-Parallel Graphs

Abstract

Given a graph G = (V, E), where V and E are vertex and edge sets of G, and a subset V NT of vertices called a non-terminal set, the minimum spanning tree with a non-terminal set V NT , 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.

A Polynomial-Time Algorithm for Finding a Spanning Tree with Non-Terminal Set <i>V_NT</i> on Circular-Arc Graphs

“…Zhang and Yin [14] showed that the problem for finding an MSTNT is NP-hard and described an approximation algorithm for finding an MSTNT on general graphs. We developed a linear-time algorithm on an outerplanar graph [11] and on a series-parallel graph [13]. This problem can be applied to the design of computer networks where a devise used for a non-terminal node has a function just as a relay point while a devise used for a terminal node has a function of an interface in addition to a relay point.…”