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...

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123–129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs: an important one is that they are strongly connected. In [E. Cheng, M.J. Lipman, On the Day–Tripathi orientation of the star graphs: Connectivity, Inform. Process....

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in 2-connected or 2-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if G is a 2-connected k-regular graph, k ≥ 3, then every edge ...

Several authors have studied the graphs for which every edge is a chord of a cycle; among 2-connected graphs, one characterization is that the deletion of one vertex never creates a cut-edge. Two new results: among 3-connected graphs with minimum degree at least 4, every two adjacent edges are chords of a common cycle if and only if deleting two vertices never creates two adjacent cut-edges; am...

We present the rst truly polynomial-time approximation scheme (PTAS) for the minimum-cost k-vertex-(or, k-edge-) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph on a superset of the input points which is k-vertex connected with respect to the input points, and whose cos...

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai [Z.-H. Chen, H.-J. Lai, Reduction techniques for super-Eulerian graphs and related topics— an update, in: Ku Tung-Hsin (Ed.), Combinatorics and Graph Theory, vol. 95, World Scientific, Singapore/London, 1995, pp. 53–69, Conjecture 8.6] conjectured that every 3-edge connected, essentially 6-edge connected graph ...