Structural properties of recursively partitionable graphs with connectivity 2

Olivier Baudon (1) , Julien Bensmail (1, 2) , Florent Foucaud (3) , Monika Pilsniak (4)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
3 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
4 AGH UST - AGH University of Science and Technology [Krakow, PL]

A connected graph G is said to be arbitrarily partitionable (AP for short) if for every partition ($n_1, ..., n_p$) of |V(G)| there exists a partition ($V_1, ..., V_p$) of V(G) such that each V_i induces a connected subgraph of G on $n_i$ vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of G must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.

Mots clés: structural properties; balloon; recursively arbitrarily partitionable graph; balloon graph; graph with connectivity 2; online arbitrarily partitionable graph

Domaines: Mathématique discrète [cs.DM]

Olivier Baudon, Julien Bensmail, Florent Foucaud, Monika Pilsniak. Structural properties of recursively partitionable graphs with connectivity 2. Discussiones Mathematicae Graph Theory, 2017, 37, pp.89-115. ⟨10.7151/dmgt.1925⟩. ⟨hal-00672505v4⟩