2-walks in 3-connected Planar Graphs

Erratum to 2-walks in 3-connected planar graphs

Enumerative Properties of Rooted Circuit Maps

In 1966 Barnette introduced a set of graphs, called circuit graphs, which are obtained from 3-connected planar graphs by deleting a vertex. Circuit graphs and 3-connected planar graphs share many interesting properties which are not satisfied by general 2-connected planar graphs. Circuit graphs have nice closure properties which make them easier to deal with than 3-connected planar graphs for s...

Decomposing Infinite 2-Connected Graphs into 3-Connected Components

In the 1960’s, Tutte presented a decomposition of a 2-connected finite graph into 3-connected graphs, cycles and bonds. This decomposition has been used to reduce problems on 2-connected graphs to problems on 3-connected graphs. Motivated by a problem concerning accumulation points of infinite planar graphs, we generalize Tutte’s decomposition to include all infinite 2-connected graphs.

Series-parallel subgraphs of planar graphs

In this paper we show that every 3-connected (3-edge-connected) planar graph contains a 2-connected (respectively, 2-edge-connected) spanning partial 2-tree (seriesparallel) graph. In contrast, a recent result by [4] implies that not all 3-connected graphs contain 2-edge-connected series-parallel spanning subgraphs.

One-way infinite 2-walks in planar graphs

We prove that every 3-connected 2-indivisible infinite planar graph has a 1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2-walk is a spanning walk using every vertex at most twice.) This improves a result of Timar, which assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also provide other r...