A Locally Connected Spanning Tree Can Be Found in Polynomial Time on Simple Clique 3-Trees

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that for each node its neighborhood in T induces a connected subgraph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a linear time algorithm that, given an SC 3-tree (i.e. a maximal planar chordal graph), determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an analogous result even for the case when the input graph is an SC 2-tree (i.e. a maximal outerplanar graph).