A generalization of the block decomposition for k-connected graphs - Malpartida, Jared León (2022)
The decomposition of a connected graph by the set of its cut-vertices, sometimes called the \"block decomposition\" or \"block tree\" of a graph, is a well known and basic concept in graph theory. This decomposition, however, does not provide meaningful information when applied to a k-connected graph for k > 1. There has been a number of attempts to generalize the construction of the block decomposition of a graph for the case of k-connected graphs. Notably, Tutte constructed a tree that describes the mutual arrangement of 2-cutsets in a 2-connected graph. This decomposition has some similarities to the block decomposition of a connected graph. In other works, a block of a k-connected graph was defined as a maximal (k+1)-connected subgraph. Karpov described the decomposition of a k-connected graph by the set of k-cutsets that are not separated by any other k-cutset of the graph. Karpov also described some special properties of his decomposition for the case of a 2-connected graph. The decompositions defined by Karpov and Tutte for the case of a 2-connected graph share some similarities. In this work, we present a self-contained description of Karpov\'s decomposition. We also present some applications to the study of planarity, the chromatic number, critically 2-connected graphs, and the partition of certain 2-connected graphs into three connected subgraphs.