Partitioning vertices and edges of graphs into connected subgraphs

Abstract

Arbitrarily partitionable (AP) graphs are graphs that can have their vertices partitioned into arbitrarily many parts inducing connected graphs of arbitrary orders. Since their introduction, several aspects of AP graphs have been investigated in literature, including structural and algorithmic aspects, their connections with other fundamental notions of graph theory, and variants of the original notion. Quite recently, an edge version of AP graphs, called arbitrarily edge-partitionable (AEP) graphs have been introduced and studied, with a special focus on their similarities and discrepancies with AP graphs.

In this work, we introduce and study a total variant of AP graphs, called arbitrarily total-partitionable (ATP) graphs, which essentially stand as a combination of AP and AEP graphs, for some particular notion of connectivity for sets of vertices and edges. We establish results of several natures, which we compare to known, similar results for AP and AEP graphs. In particular, we prove that, although the involved definitions are rather close, being AP, AEP, and/or ATP for a graph does not guarantee it also has the other properties. We also establish that deciding whether a tree can be partitioned in this total way is NP-complete in general, and provide sufficient conditions for ATPness in terms of longest paths. We finally raise directions for further work on the topic.