Some basic observations on Kelly's conjecture for graphs

Manvel, Bennet

doi:10.1016/0012-365x(74)90064-8

Cited by 21 publications

(26 citation statements)

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“…The conjecture has been verified by Giles (1976b, c) for trees (see also Spencer, 1969) and, in the case k = 2, by Manvel (1974a) for disconnected graphs (except when there are exactly two components, one of which is isomorphic to K , ) .…”

Section: Reconstruction Of Graphs From Other Informationmentioning

confidence: 77%

“…It appears, however, that all sufficiently large graphs are so determined, and Manvel (1969a) has proposed the following conjecture: given any posiliue integer k, there exists an integer f(k) such that any graph on at least f(k) vertices is reconstructible from its collection of k-uertex-deleted subgraphs. Nydl (1976) constructs examples which show that f( k ) L 2 k -I (if such a function indeed exists), thereby extending an earlier lower bound due to Manvel (1974a). On the other hand, Miiller (1976) proves that, for every c>O, almost all graphs on n vertices are reconstructible from their…”

Section: Reconstruction Of Graphs From Other Informationmentioning

confidence: 92%

See 1 more Smart Citation

Graph reconstruction—a survey

Bondy

Hemminger

1977

Journal of Graph Theory

222

127

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The Reconstruction Conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. This article reviews the progress made on the conjecture since it was first formulated in 1941 and discusses a number of related questions.The Reconstruction Conjecture is generally regarded as one of the foremost unsol;ed problems in graph theory. Indeed, Harary (1969) has even classified it as a "graphical disease" because of its contagious nature. According to reliable sources, it was discovered in Wisconsin in 1941 by Kelly and Ulam, and claimed its first victim (P. J. Kelly) in 1942.* There are now more than sixty recorded cases, and relapses occur frequently (this article being a case in point). Our purpose here is to describe and analyse the current status of the disease, identify its more interesting variants, and suggest possible remedies.We shall, €or the most part, use the terminology and notation of Bondy and Murty;? so a graph G has vertex set V(G), edge set E ( G ) , v ( G ) vertices and E ( G ) edges. A subgraph of G obtained by deleting a vertex v together with its incident edges will be referred to as a vertex-deleted subgraph and denoted by G, (rather than G -v ) . Figure 1 exhibits the vertex-deleted subgraphs of a graph.

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Section: Reconstruction Of Graphs From Other Informationmentioning

confidence: 77%

Section: Reconstruction Of Graphs From Other Informationmentioning

confidence: 92%

Graph reconstruction—a survey

Bondy

Hemminger

1977

Journal of Graph Theory

222

127

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“…For sharpness, Manvel showed that the maximum degree itself is not always determined by

D_{Δ (G) + 1} (G)

. He constructed graphs

G

and

H

such that

Δ (G) = k

Δ (H) = k + 1

, and

D_{k + 1} (G) = D_{k + 1} (H)

.…”

Section: Reconstructibility Of Connectednessmentioning

confidence: 99%

“…The

2

‐deck of

G

determines only

∣ E (G) ∣

and

∣ V (G) ∣

, but the

3

‐deck determines the number of edge incidences and whether

G

is complete multipartite. At the other end, Manvel proved that for

∣ V (G) ∣ = n \geq 6

, the

(n - 2)

‐deck determines whether

G

is connected, acyclic, unicyclic, regular, or bipartite. In Section , we prove a weak version of Kelly's conjecture for connectedness: for fixed

ℓ

, when

n

is sufficiently large the

(n - ℓ)

‐deck determines whether an

n

‐vertex graph is connected.…”

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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“…Kelly [Kel57] was the first to look in this direction, Manvel [Man74] made some observations on this problem, and Bondy [Bon91, Section 11.2] surveyed related results. See also Nýdl's review [Nýd01] of the progress made on this problem in the past three decades.…”

Section: Our Contributionsmentioning

confidence: 99%