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

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 an outerplanar graph then finding an MSTNT of G is linearly solvable with respect to the number of vertices.