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.

It is known that the edge set of a 2-edge-connected 3-regular graph can be decomposed into paths of length 3. W. Li asked whether the edge set of every 2-edge-connected graph can be decomposed into paths of length at least 3. The graphs C3, C4, C5, and K4 − e have no such decompositions. We construct an infinite sequence {Fi}∞i=0 of nondecomposable graphs. On the other hand, we prove that every...

Gain graphs are graphs in which each edge has a gain (a label from a group, say ?, so that reversing the direction, inverses the gain). In this paper we take a generalized view of gain graphs in which the label of an edge is related to the label of the reverse edge by an involution. Applying selectors to gain graphs is a generalization of vertex-switching (or Seidel switching) of an undirected ...

A block in a graph G is a maximal subgraph with no cut-vertex. Blocks are either 2-connected graphs, or the graph with two vertices and only one edge, or, in the case where G is just a single vertex, G itself. Definition 1.1. Let C be a class of connected graphs, and B ⊆ C the class of 2-connected graphs in C (B may contain the graph formed by an edge). We call C stable if it satisfies the foll...