Graph reconstruction numbers

The Reconstruction Conjecture is one of the most important open problems in graph theory today. Proposed in 1942, the conjecture posits that every simple, finite, undirected graph with more than three vertices can be uniquely reconstructed up to isomorphism given the multiset of subgraphs produced by deleting each vertex of the original graph. Related to the Reconstruction Conjecture, reconstruction numbers concern the minimum number of vertex deleted subgraphs required to uniquely identify a graph up to isomorphism. During the summer of 2004, Jennifer Baldwin completed an MS project regarding reconstruction numbers. In it, she calculated reconstruction numbers for all graphs G where 2 \u3c |V(G)| \u3c 9. This project expands the computation of reconstruction numbers up to all graphs with ten vertices and a specific class of graphs with eleven vertices. Whereas Jennifer\u27s project focused on a statistical analysis of reconstruction number results, we instead focus on theorizing the causes of high reconstruction numbers. Accordingly, this project establishes the reasons behind all high existential reconstruction numbers identified within the set of all graphs G where 2 \u3c |V(G)| \u3c 11 and identifies new classes of graphs that have large reconstruction numbers. Finally, we consider 2-reconstructibility - the ability to reconstruct a graph G from the multiset of subgraphs produced by deleting each combination of 2 vertices from G. The 2-reconstructibility of all graphs with nine or less vertices was tested, identifying two graphs in this range with five vertices as the highest order graphs that are 2-nonreconstructible

A computational investigation of graph reconstruction

First proposed in 1941 by Kelly and Ulam, the Graph Reconstruction Conjecture has been called the major open problem in the field of Graph Theory. While the Graph Reconstruction Conjecture is still unproven it has spawned a number of related questions. In the classical vertex graph reconstruction number problem a vertex is deleted in every possible way from a graph G, and then it can be asked how many (both minimum and maximum values) of these subgraphs are required to uniquely reconstruct G (up to isomorphism). This problem can then be extended to k-vertex deletion (for 1 ≤ k ≤ |V (G)|), and to k-edge deletion (for 1 ≤ k ≤ |E(G)|). For some classes of graphs there is known a formula to directly compute its reconstruction numbers. However, for the vast majority of graphs the computation devolves to brute force exhaustive search. Previous computer searches have computed the 1-vertex-deletion reconstruction numbers of all graphs of up to 10 vertices, as well as computing 2-vertex-deletion reconstructibility of all graphs on up to 9 vertices. In this project I have developed and implemented an improved algorithm to compute 1-vertex-deletion reconstruction numbers with an O(|V (G)|) speedup, allowing their computation for all graphs of up to 11 vertices. In addition the ability to compute arbitrary k-vertex and edge deletion reconstruction numbers has been implemented, leading to many new results in these areas

Reconstructing trees from small cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. In 1976, Giles proved that any tree on $n\geq 6$ vertices can be reconstructed from its $\ell$-deck for $\ell \geq n-2$. Our main theorem states that it is enough to have $\ell\geq (8/9+o(1))n$, making substantial progress towards a conjecture of N\'ydl from 1990. In addition, we can recognise connectedness from the $\ell$-deck if $\ell\geq 9n/10$, and reconstruct the degree sequence from the $\ell$-deck if $\ell\ge \sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.Comment: 24 pages, fixed several typo

Reconstructing pedigrees: some identifiability questions for a recombination-mutation model

Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees - Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.Comment: 40 pages, 9 figure

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

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 be reconstructed from any deck missing O(n/d3) 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

Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer $h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set $V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever ${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}. Let $s(h)$ be the least integer $k$ such that the conclusion above holds and let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)= h+2^{\lfloor \log_2 h\rfloor}$. In the special case $h=2^{\ell}$ or $h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and $v(2)=6$ were obtained in a previous work.Comment: 15 page

Discussion On Some Important Topics in Graph Theory

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