Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices

Abstract

The k-deck of a graph is the multiset of its subgraphs induced by k vertices. A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. We show that the degree list of an n-vertex graph is 3-reconstructible when n ≥ 7, and the threshold on n is sharp. Using this result, we show that when n ≥ 7 the (n − 3)-deck also determines whether an n-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are 2-reconstructible when n ≥ 6, which are also sharp.MSC Codes: 05C60, 05C07

Acyclic graphs with at least $2\ell+1$ vertices are $\ell$-recognizable

“…This threshold is surely too large. Manvel [9] proved that connectedness is 2-recognizable for graphs with at least six vertices, and the present authors [7] proved that connectedness is 3-recognizable for graphs with at least seven vertices. Spinoza and West [11] suggested that (except for (n, ℓ) = (5, 2)), connectedness is recognizable for n-vertex graphs when n ≥ 2ℓ+1.…”

3-Regular Graphs Are 2-Reconstructible

“…Chernyak [2] showed that the degree list is 2-reconstructible when n ≥ 6 (sharp by {C 4 +K 1 , K ′ 1,3 }). The present authors [6] showed that the degree list is 3-reconstructible when n ≥ 7 (sharp by {C 5 + K 1 , K ′′ 1,3 }, where K ′′ 1,3 is the tree obtained from K 1,3 by subdividing two edges). For ℓ in general, Taylor [11] showed that the degree list is ℓ-reconstructible when n ≥ eℓ + O(log ℓ), where e is the base of the natural logarithm.…”

Reconstructing trees from small cards

“…We have already mentioned Manvel's result in [18] that the class of connected graphs is recognisable from the (n − 2)-deck for n ≥ 6. Extending this, Kostochka, Nahvi, West, and Zirlin [14] showed that the connectedness of a graph on n ≥ 7 vertices is determined by D n−3 (G). As shown by Spinoza and West [22], if we take G 1 = P n (the path on n vertices) and G 2 = C ⌈n/2⌉+1 ⊔ P ⌊n/2⌋−1 the disjoint union of a cycle and a path, we find…”

“…Suppose ℓ is an integer such that ℓ ≥ 9n/10 = (1 − ε)n. We wish to recognise if G is connected from the ℓ deck. It was shown by Kostochka, Nahvi, West, and Zirlin [14] that the connectedness of a graph can be recognised from the (n − 3)-deck for n ≥ 7, so we can assume that n ≥ 39. Suppose that G is disconnected and let H be the largest component.…”

“…The story in the literature here is similar to that for connectedness. Chernyak [7] showed that the degree sequence of an n-vertex graph can be reconstructed from its (n − 2)-deck for n ≥ 6, and this was later extended by Kostochka, Nahvi, West, and Zirlin [14] to the (n − 3)-deck for n ≥ 7. The best known asymptotic result is due to Taylor [23], and implies that the degree sequence of a graph G on n vertices can be reconstructed from D ℓ (G) where ℓ ∼ (1 − 1/e)n. Our improved bound is as follows.…”