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Kelly–Ulam Conjecture
Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to KellyKelly, P. J.A congruence theorem for trees ''Pacific J. Math.'' 7 (1957), 961–968. and Ulam.Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960. Formal statements Given a graph G = (V,E), a vertex-deleted subgraph of G is a subgraph formed by deleting exactly one vertex from G. By definition, it is an induced subgraph of G. For a graph G, the deck of G, denoted D(G), is the multiset of isomorphism classes of all vertex-deleted subgraphs of G. Each graph in D(G) is called a card. Two graphs that have the same deck are said to be hypomorphic. With these definitions, the conjecture can be stated as: * Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic. : (The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
