https://hal.archives-ouvertes.fr/hal-00687278/document

Partitioning powers of traceable or hamiltonian graphs

On the structure of arbitrarily partitionable graphs with given connectivity

Abstract A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with ‖G‖ > (n−42)+12 $||G||\\; > \\;\\left( {\\matrix{{n - 4} \\cr 2 \\cr } } \\right) + 12$ edges is AP or belongs to few classes of exceptional graphs.

/pdf/dense-arbitrarily-partitionable-graphs-3uv0eoufit.pdf

Dense Arbitrarily Partitionable Graphs

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.

https://hal.archives-ouvertes.fr/hal-00672505/document

Structural properties of recursively partitionable graphs with connectivity 2

A graph G of order n is arbitrarily partitionable (AP) if, for every sequence (n1,...,np) partitioning n, there is a partition (V1,...,Vp) of V (G) such that G[Vi] is a connected graph of order ni for i=1,...,p. The property of being AP is related to other well-known graph notions, such as perfect matchings and Hamiltonian cycles, with which it shares several properties. This work is dedicated to studying two aspects behind AP graphs. 

On the one hand, we consider algorithmic aspects of AP graphs, which received some attention in previous works. We first establish the NP-hardness of the problem of partitioning a graph into connected subgraphs following a given sequence, for various new graph classes of interest. We then prove that the problem of deciding whether a graph is AP is in NP for several classes of graphs, confirming a conjecture of Barth and Fournier for these. 

On the other hand, we consider the weakening to APness of sufficient conditions for Hamiltonicity. While previous works have suggested that such conditions can sometimes indeed be weakened, we here point out cases where this is not true. This is done by considering conditions for Hamiltonicity involving squares of graphs, and claw-and net-free graphs.

pdf/more-aspects-of-arbitrarily-partitionable-graphs-1clkpiav1t.pdf

More Aspects of Arbitrarily Partitionable Graphs

A graph G = (V,E) is arbitrarily vertex decomposable if for any sequence tau of positive integers adding up to |V |, there is a sequence of vertex-disjoint subsets of V whose orders are given by tau, and which induce connected graphs. The main aim of this paper is to study the recursive version of this problem. We present a solution for trees, suns, and partially for a class of 2-connected graphs called balloons.

/pdf/recursively-arbitrarily-vertex-decomposable-graphs-1ad1adqkvt.pdf

Recursively arbitrarily vertex-decomposable graphs

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Recursively arbitrarily vertex-decomposable graphs