3-Regular Graphs Are 2-Reconstructible

Kostochka, Alexandr; Nahvi, Mina; West, Douglas B.; Zirlin, Dara

doi:10.48550/arxiv.1908.01258

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2021

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“…Manvel showed that several classes of graphs, such as connected graphs, trees, regular graphs and bipartite graphs, can be recognised from the (n − 2)deck where n ≥ 6 is the number of vertices -that is, whether a given graph is a member of such a class is determined by its (n − 2)-deck. Since then, the problem has been widely studied, and the reconstructibility of graphs from smaller cards is known for many classes of graphs including trees, 3-regular graphs, random graphs and graphs with maximum degree 2 [8,13,22]. However, many of the bounds are far from tight.…”

Section: Introductionmentioning

confidence: 99%

Reconstructing trees from small cards

Groenland,

Johnston,

Scott

et al. 2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%