A generalization of the block decomposition for k-connected graphs - León, Jared (Anais do Encontro de Teoria da Computação (ETC); 2022: Anais do VII Encontro de Teoria da ComputaçãoDO - 10.5753/etc.2022.223106, 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 any meaningful information when applied to a $k$-connected graph for $k ≥ 2$. 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. In this work, we present an outline of one such attempt by Karpov. We also present some applications of this decomposition to the study of planarity, the chromatic number, and critically $2$-connected graphs.