What is a minimum spanning tree?

It's a subset of a graph's edges that connects every node together, contains no cycles, and has the minimum possible total edge weight among all such connecting subsets.

How does Kruskal's algorithm work?

It sorts all edges by weight from smallest to largest, then processes them in order, adding each edge to the tree only if it connects two previously-disconnected parts of the graph (checked efficiently with a union-find structure).

What is union-find used for?

Union-find (also called disjoint-set) efficiently tracks which nodes are already connected, letting Kruskal's algorithm quickly detect whether adding an edge would create a cycle, without re-scanning the whole tree each time.

What if my graph is disconnected?

The calculator reports a minimum spanning forest instead — the cheapest way to connect each disconnected component internally, since no single tree can span nodes that have no path between them.

Is the minimum spanning tree always unique?

Not necessarily — if multiple edges share the same weight, there can be more than one valid minimum spanning tree with the same total weight, though the total weight itself is always unique.

How does this differ from Dijkstra's algorithm?

A minimum spanning tree minimizes the total cost to connect all nodes together, while Dijkstra's algorithm minimizes the cost of a path between two specific nodes — the two problems solve different questions even though both work on weighted graphs.

How many nodes can this calculator handle?

This calculator supports up to 50 nodes, more than enough for hand-entered graph examples while keeping the sort-and-union-find computation instant.

Where are minimum spanning trees used in real life?

They're used to design cost-efficient networks — laying cable or pipeline routes, wiring circuit boards, clustering data points, and approximating solutions to harder network-design problems.

Minimum Spanning Tree Calculator — Find the Cheapest Way to Connect a Graph

Our Minimum Spanning Tree Calculator finds the set of edges that connects every node in a weighted graph at the lowest possible total cost, using Kruskal's algorithm with union-find cycle detection, given your graph as a simple text edge list (e.g. A-B:4, B-C:2).

Quick Answer

Enter your weighted graph as an edge list to instantly find the minimum spanning tree — the cheapest set of edges connecting every node — via Kruskal's algorithm.

Edges (comma-separated)

Enter weighted edges as NodeA-NodeB:weight, e.g. A-B:4, B-C:2.

How to Use the Minimum Spanning Tree Calculator — Kruskal's Algorithm Online

1
Enter your graph as a comma-separated edge list, e.g. A-B:4, B-C:2.
2
Click 'Calculate' to find the minimum spanning tree.
3
Review the selected edges and the total minimum weight.

Why Use Minimum Spanning Tree Calculator — Kruskal's Algorithm Online?

A minimum spanning tree connects every node in a graph using the fewest and cheapest possible edges, with no cycles — exactly the structure you'd want when designing a network (roads, pipelines, circuit wiring) that must reach every point at minimum total cost. Kruskal's algorithm finds it elegantly: sort all edges from cheapest to most expensive, and greedily add each edge that doesn't create a cycle, tracking connectivity efficiently with a union-find data structure. This greedy approach is provably optimal, and this calculator runs it directly on a graph you enter as a weighted edge list.

Frequently Asked Questions

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