On the reconstruction of planar graphs

Bilinski, Mark; Kwon, Young Soo; Yu, Xingxing

doi:10.1016/j.jctb.2006.12.005

Cited by 12 publications

(5 citation statements)

References 10 publications

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“…Indeed, consider adding disjoint copies of the same (3p + 1)-vertex graph H to both G 1 and G 2 . The resulting graphs will still have about 2 3 × 1 2 = 1 3 of their cards in common, and we can create the desired average degree by choosing the density of H.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, our results hold for planar graphs. A graph class C is recognisable if no graph G ∈ C has the same deck of cards as any graph H ∈ C. Planar graphs are of particular interest as they were shown to be recognisable by Bilinksi, Kwan and Yu in [2], but are still not known to be reconstructible in general. Partial results include the reconstructibility of outerplanar graphs [8], maximal planar graphs [12], and certain 5-connected planar graphs [2].…”

Section: Introductionmentioning

confidence: 99%

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Reconstructing the degree sequence of a sparse graph from a partial deck

Groenland,

Johnston,

Kupavskii

et al. 2021

Preprint

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The deck of a graph G is the multiset of cards {G − v : v ∈ V (G)}. Myrvold (1992) showed that the degree sequence of a graph on n ≥ 7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can reconstructed from any deck missing O(n/d 3 ) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reconstructing the degree sequence of a sparse graph from a partial deck

Groenland,

Johnston,

Kupavskii

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, there are classes of graphs that are recognizable, but not known to be reconstructible. An important example is the class of planar graphs (Bilinski et al [1]).…”

Section: Definition 24 a Class Of Graphs Is A Set Of Graphs That Ismentioning

confidence: 99%

An Algebraic Formulation of the Graph Reconstruction Conjecture

Oliveira

Thatte

2015

Journal of Graph Theory

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Abstract:The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck-the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph G and any finite sequence of graphs, it gives a linear constraint that every reconstruction of G must satisfy. Let ψ (n) be the number of distinct (mutually nonisomorphic) graphs on n vertices, and let d (n) be the number of distinct decks that can be constructed from these graphs. Then the difference ψ (n) − d (n) measures how many graphs cannot be reconstructed from their decks. In particular, the graph We give a framework based on Kocay's lemma to study this discrepancy. We prove that if M is a matrix of covering numbers of graphs by sequences of graphs, then d (n) ≥ rank R (M). In particular, all n-vertex graphs are reconstructible if one such matrix has rank ψ (n). To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix M of covering numbers satisfies d (n) = rank R (M).

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“…Every finite simple graph with at least three vertices will be reconstructible? Some classes of reconstructible graphs are known (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and [13]) as follows: Let Γ be a finite simple graph with at least three vertices.…”

Section: Introductionmentioning

confidence: 99%

The Reconstruction Conjecture for finite simple graphs and associated directed graphs

Hosaka¹

2020

Preprint

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In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and Γ be finite simple graphs with at least three vertices such that there exists a bijective map f : V (Γ) → V (Γ ) and for any v ∈ V (Γ), there exists an isomorphism φv : Γ − v → Γ − f (v). Then we define the associated directed graph Γ = Γ(Γ, Γ , f, {φv} v∈V (Γ) ) with two kinds of arrows from the graphs Γ and Γ , the bijective map f and the isomorphisms {φv} v∈V (Γ) . By investigating the associated directed graph Γ, we study when are the two graphs Γ and Γ isomorphic. In particular, we show that if Γ and Γ do not have some structure then they are isomorphic.

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On the reconstruction of planar graphs

Abstract: We show that the planarity of a graph can be recognized from its vertex deleted subgraphs, which answers a question posed by Bondy and Hemminger in 1979. We also state some useful counting lemmas and use them to reconstruct certain planar graphs.

Cited by 12 publications

References 10 publications

Reconstructing the degree sequence of a sparse graph from a partial deck

Reconstructing the degree sequence of a sparse graph from a partial deck

An Algebraic Formulation of the Graph Reconstruction Conjecture

The Reconstruction Conjecture for finite simple graphs and associated directed graphs

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