On minimal arbitrarily partitionable graphs

Interplays between variations of arbitrarily partitionable graphs under minimality constraints

“…• In [5], Baudon, Przybyło, and Woźniak exhibited a construction for a minAP graph having arbitrarily many cycles, thus being arbitrarily far from trees. We were not able to exhibit similar constructions for minOLAP and minRAP graphs, and we believe determining whether such graphs exist would be crucial for this field.…”

“…• For λ = 1, we consider S = {h}. Then G[S] is the 1-graph, which is RAP, while G − S is spanned by Cat (3,5), which is RAP by Theorem 2.3.…”

“…This led to the definition of AP graphs that are minimal, which we call minAP throughout, in the sense that they are not spanned by any non-trivial AP graph (i.e., by an AP graph with strictly less edges). Later on, Baudon, Przybyło, and Woźniak, in [5], came up with a construction showing that a minAP n-graph can have its number of edges being arbitrarily larger than n − 1, the size of a tree on n vertices. Still, it is believed that minAP n-graphs should be somewhat sparse, meeting a conjecture of Ravaux from [20] stating they should have linear size (of order O(n)).…”

Arbitrarily partitionable \{2K2, C4\}-free graphs

“…It follows that (|A 1 |, |A 2 |, |A 3 |, |A 4 |, |A 5 |) = (1, 1, 1, 3, 3) or (1, 2, 1, 3, 2) or (1, 3, 1, 2, 2). Since λ 1 ≤ 9 3 , for the cases when (1, 1, 1, 3, 3) and (1, 2, 1, 3, 2), we can obtain G 1 from G by deleting λ 1 vertices from A 4 ∪ A 5 . For the case when (1, 3, 1, 2, 2), if λ 1 ≤ 2, we can obtain G 1 from G by deleting λ 1 vertices from A 4 ∪ A 5 .…”

“…Structure of AP graphs and minimal AP graphs are investigated in [7,9]. The problem of deciding whether a given admissible sequence is realizable in a given graph G is NP-complete [2].…”