A Survey on Algorithms for the Maximum Internal Spanning Tree and Related Problems

Given an undirected connected graph G we consider the problem of finding a spanning tree of G with a maximum number of internal ( ⩾ 2 degree) vertices. This problem, called the Maximum Internal Spanning Tree problem, is obviously NP-hard since it generalizes the Hamiltonian Path problem. In this paper we aim at giving a survey on recent results about the Maximum Internal Spanning Tree problem including different approaches such as exact exponential algorithms, fixed parameter tractability, and approximation algorithms. We also consider the problem of finding a large ( ⩽ q )-leaf subtree of the input graph for some fixed q.