Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an $O(|V|^2)$ algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an $O(|V|^3)$ algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.

In this paper, we describe an O(|V(G)||E(G)|) algorithm for finding a nonseparating planar chain in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into planar chains. This work was motivated by the study of a multitree approach to reliability in distributed networks, as well as the study of nonseparating induced paths in highly connected graphs.

In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an $O(|V(G)|^2|E(G)|)$ algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to $O(|V(G)|^3)$.This decomposition will be used to find four independent spanning trees in a 4-connected graph.

We prove that given any fixed edge ra in a 4-connected graph G, there exists a cycle C through ra such that G-(V(C)-{r}) is 2-connected. This will provide the first step in a decomposition for 4-connected graphs. We also prove that, for any given edge e in a 5-connected graph G, there exists an induced cycle C through e in G such that G-V(C) is 2-connected. This provides evidence for a conjecture of Lovász.