Finding Nonseparating Induced Cycles and Independent Spanning Trees in 3-Connected Graphs

Abstract Given a vertex r of a 3-connected graph G , we show how to find three independent spanning trees of G rooted at r . Our proof is based on showing that every 3-connected graph has a nonseparating ear decomposition. This extends Whitney's characterisation that a graph is 2-connected iff it has an ear decomposition. We also show that a nonseparating ear decomposition can be constructed in O ( VE ) time, and hence, three independent spanning trees can be found in O ( VE ) time. We construct a nonseparating ear decomposition by solving the following problem at most V times. Given an edge tr and a vertex u of a 3-connected graph G , find a nonseparating induced cycle of G through tr and avoiding u . W. T. Tutte (Proc. London Math. Soc. 13 (1963), 743–767) first showed that such a cycle can always be found. We give a linear time algorithm for this.