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The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
In combinatorics, he is known for Kruskal's tree theorem (1960), which is also interesting from a mathematical logic perspective since it can only be proved nonconstructively. Kruskal also applied his work in linguistics , in an experimental lexicostatistical study of Indo-European languages , together with the linguists Isidore Dyen and Paul ...
The article as written is mostly about the TREE() function, and describes barely more of Kruskal's tree theorem than is needed to explain the TREE() function's terminology. Even the subsection describing the weak tree() function fails to explain who defined it or for what purpose; is it any more relevant to Kruskal's trees than the rest of this? G.
Then any maximum-weight spanning tree of the clique graph is a junction tree. So, to construct a junction tree we just have to extract a maximum weight spanning tree out of the clique graph. This can be efficiently done by, for example, modifying Kruskal's algorithm. The last step is to apply belief propagation to the obtained junction tree. [10]
Chapter nine discusses ways to weaken Ramsey's theorem, [2] and the final chapter discusses stronger theorems in combinatorics including the Dushnik–Miller theorem on self-embedding of infinite linear orderings, Kruskal's tree theorem, Laver's theorem on order embedding of countable linear orders, and Hindman's theorem on IP sets. [3]
Kruskal's tree theorem states that, in every infinite set of finite trees, there exists a pair of trees one of which is homeomorphically embedded into the other; another way of stating the same fact is that the homeomorphisms of trees form a well-quasi-ordering.