Reconstructing trees from small cards

Abstract

The ℓ-deck of a graph G is the multiset of all induced subgraphs of G on ℓ vertices. In 1976, Giles proved that any tree on n ≥ 6 vertices can be reconstructed from its ℓ-deck for ℓ ≥ n − 2. Our main theorem states that it is enough to have ℓ ≥ (8/9 + o(1))n, making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise connectedness from the ℓ-deck if ℓ ≥ 9n/10, and reconstruct the degree sequence from the ℓ-deck if ℓ ≥ 2n log(2n). All of these results are significant improvements on previous bounds.

Optimally reconstructing caterpillars

“…We will actually need very little notation about graphs. We will use just a few notations from [1]. The diameter of graph G is the maximum distance between two vertices u, v ∈ V (G), when G is a tree this is the same as the number of edges in the longest path in G. We let n H (G) count the number of times a certain graph H appears as a subgraph of G (i.e.…”

“…Remark 2.2. We can reconstruct the degree sequence of G from D √ 2n log(2n) (G), see Theorem 7 of [1].…”

“…(1) confirm G is a caterpillar graph, and recognize its diameter, (2) use D ℓ (G) to reconstruct D S (ϕ(G)) for some appropriate sets S, (3) use the S-decks D S (ϕ(G)) to reconstruct ϕ(G). Because caterpillars are quite structured, step (1) becomes trivial thanks to results from recent literature [1,2] (this is covered at the start of Section 4). This leaves steps (2) and (3).…”

“…In Section 3, we establish a key lemma about equalities between "moments". It is essentially a multivariable analogue of [1,Lemma 11]. Our assumptions are a bit stronger, which allows for an elementary number theoretic proof.…”

“…Then in Subsection 2.1 we outline our reconstruction procedure, which will be divided into three parts. At its heart, our methods are a more precise utilization of H-extensions (a concept introduced in [1, Section 5.1]), where we figured out how to efficiently reconstruct a special class of H-extensions and then realized how to reconstruct caterpillars with this special class (which carried much less information than the H-extensions considered in [1]).…”