# A note on two problems in connexion with graphs

TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.

Abstract: 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 branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

## Summary (1 min read)

### Summary

- Problem 1. Construct 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 ol the construction that the authors present here, the branches are subdivided into three sets: I. the branches definitely assigned to the tree under construction (they will form a subtree) ; IL the branches from which the next branch to be added to set I, will be selected; III.
- The rernaining branches (rejected or not yet considered)'.
- The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes).
- From then onwalals the authors Pedorm the following two steps repeatedly.

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##### Citations

9,164 citations

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### Cites methods from "A note on two problems in connexion..."

...A simpler technique that is computationally tractable is to take a dynamic programming approach to the calculation of distances and employ an algorithm typically used to calculate minimal distances in a graph ( Dijkstra, 1959 )....

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7,219 citations

7,172 citations

### Cites background from "A note on two problems in connexion..."

...Dealing with large graphs can be different though – if it takes three months to calculate the diameter of a graph, nobody wants that to be interactive....

[...]

5,623 citations

### Cites methods from "A note on two problems in connexion..."

...The result is the well-known Dijkstra’s algorithm for finding single-source shortest paths in a graph [275], which is a special form of dynamic programming....

[...]

##### References

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