140,521 research outputs found
An algebraic formulation of the graph reconstruction conjecture
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
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on
vertices, and let be the number of distinct decks that can be
constructed from these graphs. Then the difference measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for -vertex graphs if and only if
.
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if is a matrix of covering numbers of graphs by sequences of
graphs, then . In particular, all
-vertex graphs are reconstructible if one such matrix has rank . To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix of covering numbers
satisfies .Comment: 12 pages, 2 figure
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
Graph reconstruction numbers
Proposed in 1942, the Graph Reconstruction Conjecture posits that every simple, finite, undirected graph with three or more vertices can be reconstructed up to isomorphism to the original graph, given the multiset of subgraphs produced by deleting each vertex along with its incident edges. Related to this Reconstruction Conjecture, existential reconstruction numbers, 9rn(G), concern the minimum number of vertex-deleted subgraphs required to identify a graph up to isomorphism. We discuss the resulting data from calculating reconstruction numbers for all simple, undirected graphs with up to ten vertices. From this data, we establish the reasons behind all high existential reconstruction numbers (9rn(G) \u3e 3) for |V (G)| is less than or equal to 10 and identify new classes of graphs that have high reconstruction numbers for |V (G)| \u3e 10. We also consider 2-reconstructibility { the ability to reconstruct a graph G from the multiset of subgraphs produced by deleting each combination of two vertices from G. The 2-reconstructibility of all graphs with nine or less vertices was tested, identifying four graphs in this range with five vertices as the highest order of graphs that are not 2-reconstructible
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
Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph
An ecard of a graph is a subgraph formed by deleting an edge. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph is the minimum number of da-ecards that uniquely determines The adversary degree associated edge reconstruction number of a graph is the minimum number such that every collection of da-ecards of uniquely determines The maximal subgraph without end vertices of a graph which is not a tree is the pruned graph of It is shown that of complete multipartite graphs and some connected graphs with regular pruned graph is or We also determine and of corona product of standard graphs
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