# A note on two problems in connexion with graphs

@article{Dijkstra1959ANO,
title={A note on two problems in connexion with graphs},
author={E. Dijkstra},
journal={Numerische Mathematik},
year={1959},
volume={1},
pages={269-271}
}
• E. Dijkstra
• Published 1959
• Mathematics, Computer Science
• Numerische Mathematik
We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the… Expand
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#### References

SHOWING 1-8 OF 8 REFERENCES
Formal Procedures for Connecting Terminals with a Minimum Total Wire Length
• Computer Science
• JACM
• 1957
Two methods for systematically selecting the shortest connections from a list of possible connections to obtain a minimum total wire length are presented, which will be called a minimum tree. Expand
On the shortest spanning subtree of a graph and the traveling salesman problem
7. A. Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull. Acad. Sei. URSS, Ser. Math. vol. 5 (1941) pp. 233-240. 8. J.Expand
NETWORK FLOW THEORY
Abstract : The labeling algorithm for the solution of maximal network flow problems and its application to various problems of the transportation type are discussed.
Mathematisch Centrum 2e Boerhaavestraat 49 Amsterdam-O (Received ]1tne 11, 19:WJ Druck tier Univ,,rsitiitstlruckerei II
• Theorie des graphes et ses applications,
• 1958
I heorie des graphes et ses applications, pp. 6g _ 69. paris: Dunod t 95g
• Mattreoatisch Centlum 2e Boerhaavestraat 49 Amsterdam-O (Recefued
• 1959
VVEINBERGER: Formal Procedures for Connecting Terminals with a Minimum Total ,vire Length
• J. Ass. Comp. Mach
• 1957
WETNaERGER : Formal procedures ior Co " nJciiog ferminals with a Minimum Total Wire Length
• J . Ass . Comp . Macb . I heorie des graphes et ses applications Mattreoatisch Centlum 2 e Boerhaavestraat 49 AmsterdamO ( Recefued