Reconstruction from the deck of ‐vertex induced subgraphs

Abstract

The k‐ deck of a graph is its multiset of subgraphs induced by k vertices; we study what can be deduced about a graph from its k‐deck. We strengthen a result of Manvel by proving for ℓ ∈ double-struckN that when n is large enough ( n > 2 ℓ ( ℓ + 1 ) 2 suffices), the ( n − ℓ )‐deck determines whether an n‐vertex graph is connected ( n ≥ 25 suffices when ℓ = 3, and n ≤ 2 l cannot suffice). The reconstructibility ρ ( G ) of a graph G with n vertices is the largest ℓ such that G is determined by its ( n − ℓ )‐deck. We generalize a result of Bollobás by showing ρ ( G ) ≥ ( 1 − o ( 1 ) ) n ∕ 2 for almost all graphs. As an upper bound on min ρ ( G ), we have ρ ( C n ) = ⌈ n ∕ 2 ⌉ = ρ ( P n ) + 1. More generally, we compute ρ ( G ) whenever normalΔ ( G ) = 2, which involves extending a result of Stanley. Finally, we show that a complete r‐partite graph is reconstructible from its ( r + 1 )‐deck.

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

“…Their full result is quite complicated to state, but a special case is that for n ≥ 2ℓ + 1 (except (n, ℓ) = (5, 2)), every n-vertex graph with maximum degree at most 2 is ℓ-reconstructible. A path with 2ℓ vertices has the same ℓ-deck as the disjoint union of an (ℓ + 1)-cycle and a path with ℓ − 1 vertices, as shown in [11], so the result of [11] and the result in the present paper are both sharp.…”

“…The two trees are obtained from a path with 2ℓ − 1 vertices by adding one leaf, either to the central vertex of the path or to one of its neighbors. Kostochka and West [8] used the results of [11] to give a short proof that these two trees have the same ℓ-deck. With our result, Nýdl's conjecture can be strengthened as follows.…”

“…Besides acyclicity, another fundamental property of trees is connectedness. Spinoza and West [11] proved that connectedness is ℓ-recognizable for n-vertex graphs when n > 2ℓ (ℓ+1) 2 . This threshold is surely too large.…”

“…For n ≥ 2ℓ + 1 (except (n, ℓ) = (5, 2)), the (n − ℓ)-deck of an n-vertex graph determines whether the graph is a tree. [11] determined for every graph G with maximum degree at most 2 the maximum ℓ such that G is ℓ-reconstructible. Their full result is quite complicated to state, but a special case is that for n ≥ 2ℓ + 1 (except (n, ℓ) = (5, 2)), every n-vertex graph with maximum degree at most 2 is ℓ-reconstructible.…”

Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

“…But this conjecture is still open in the case of symmetric digraphs. For this problem of reconstruction, we cite [3,4,38]. As for R. Fraïssé [22], he conjectured the (≤ k)-reconstruction of relations (of any arity m), k is a sufficiently large integer.…”