Recognizing connectedness from vertex-deleted subgraphs

The deck of a graph G is given by the multiset of (unlabeled) subgraphs G v v V G { − : ()} ∈. The subgraphs G v − are referred to as the cards of G. Brown and Fenner recently showed that, for n 29 ≥ , the number of edges of a graph G can be computed from any deck missing 2 cards. We show that, for sufficiently large n, the number of edges can be computed from any deck missing at most n 1 20 cards.

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“…Some parameters are reconstructible even if there is not a full deck of cards. For example, Bowler, Brown, Fenner, and Myrvold [6] showed that any

true⌊ \frac{n}{2} true⌋ + 2

cards suffice to determine whether the graph is connected. Myrvold [13] also found that the degree sequence is reconstructible from any

n - 1

cards.…”

Section: Introductionmentioning

confidence: 99%

Size reconstructibility of graphs

Groenland

Guggiari

Scott

2020

Journal of Graph Theory

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“…We can also study the reconstruction number of graph properties. Myrvold and Bowler et al showed that any

⌊ n ∕ 2 ⌋ + 2

cards determine whether an

n

‐vertex graph is connected. Woodall proved for

n \geq \max {34, 3 p^{2} + 1}

that the number of edges in an

n

‐vertex graph is reconstructible within

p - 2

from

n - p

cards; this extends and .…”

Section: Introductionmentioning

confidence: 99%

Reconstruction from the deck of ‐vertex induced subgraphs

Spinoza

West

2018

Journal of Graph Theory

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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.

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“…. Moreover, they conjectured that b(G, H) is bounded above by 2(n−1) 3 for large enough n. In a subsequent paper [6], they, together with Myrvold, showed that if G is disconnected and H is connected then b(G, H) ≤ n 2 + 1. They also characterised all pairs of such graphs that attain this bound (most of these infinite families can also be found in [5]).…”

mentioning

confidence: 99%

“…2 , when G is disconnected and H is connected (see Theorems 3.4 and 3.6 of [5] and [6], respectively.) We also show that there exist maximum saturating sets that are subsets of the automorphism group of the corresponding supercard.…”

mentioning

confidence: 99%

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A new approach to graph reconstruction using supercards

Brown

Fenner

2018

Journal of Combinatorics

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The vertex-deleted subgraph G − v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertexdeleted subgraphs. The number of common cards of G and H is the cardinality of a maximum multiset of common cards, i.e., the multiset intersection of the decks of G and H. We introduce a new approach to the study of common cards using supercards, where we define a supercard G + of G and H to be a graph that has at least one vertex-deleted subgraph isomorphic to G, and at least one isomorphic to H. We show how maximum sets of common cards of G and H correspond to certain sets of permutations of the vertices of a supercard, which we call maximum saturating sets. We then show how to construct supercards of various pairs of graphs for which there exists some maximum saturating set X contained in Aut(G + ). For certain other pairs of graphs, we show that it is possible to construct G + and a maximum saturating set X such that the elements of X that are not in Aut(G + ) are in oneto-one correspondence with a set of automorphisms of a different supercard G + λ of G and H. Our constructions cover nearly all of the published families of pairs of graphs that have a large number of common cards.

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Recognizing connectedness from vertex-deleted subgraphs

Cited by 14 publications

References 7 publications

Size reconstructibility of graphs

Size reconstructibility of graphs

Reconstruction from the deck of ‐vertex induced subgraphs

A new approach to graph reconstruction using supercards

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