Graph Theory AI Solver
Visualize & Solve Graph Problems

Graph Theory Solver — AI-Powered Step-by-Step Algorithm Solutions

This graph theory solver handles the full range of standard graph algorithm problems that appear in university computer science courses. Unlike calculators that return a final answer without explanation, every solution traces the complete execution of the algorithm — every node visited, every distance updated, every edge evaluated — so you can follow exactly what the algorithm does at each step and why.

The solver accepts graphs described as plain-text edge lists, which is the same format used in most textbooks and course notes. You don’t need to learn any special notation. Enter your edges, select the algorithm, and the solver parses the graph and produces a complete step-by-step trace in the same notation your professor would use on the board.

Shortest Path Algorithms: Dijkstra’s vs Bellman-Ford

The two main shortest path algorithms covered differ primarily in what kinds of graphs they handle and how efficient they are. Choosing the right one depends on your graph’s properties:

For most university discrete math and algorithms problems, Dijkstra’s is the expected algorithm unless negative weights are explicitly stated. The solver defaults to Dijkstra’s and will flag if Bellman-Ford is more appropriate for your input.

Minimum Spanning Trees: Kruskal’s vs Prim’s Algorithm

Both Kruskal’s and Prim’s algorithms produce a minimum spanning tree of a connected weighted graph — a tree that connects all vertices with the minimum total edge weight. They differ in their approach and which graph types they work best with.

Kruskal’s Algorithm

Kruskal’s takes a global view: it sorts all edges by weight and adds each edge in order, skipping any edge that would create a cycle. It uses a Union-Find (disjoint set union) data structure to efficiently check whether two nodes are already connected. The solver shows the sorted edge list, each Union-Find operation, and which edges are included or rejected at each step.

Prim’s Algorithm

Prim’s takes a local view: starting from a source node, it greedily expands to the nearest unvisited vertex at each step. It uses a priority queue and maintains a set of visited nodes. The solver shows the priority queue state at each step, the node selected, and which edge is added to the MST. Prim’s is typically faster on dense graphs due to fewer priority queue operations.

The key property of both algorithms is the cut property of spanning trees: at each step, both algorithms select the minimum-weight edge crossing some cut of the graph, which guarantees optimality by the matroid theory underlying MST algorithms.

Common Mistakes in Graph Theory Problems

Students solving graph algorithm problems by hand — especially on exams — consistently make a few recurring errors. Being aware of these helps you avoid them:

• In Dijkstra’s algorithm, not re-evaluating a node when a shorter path is found before it is finalized — distance updates must propagate correctly through the priority queue
• Confusing the adjacency matrix and adjacency list representations when computing algorithm complexity — the same algorithm runs in different time depending on which representation is used
• Applying Dijkstra’s to graphs with negative edge weights, where it gives incorrect results — Bellman-Ford is required for negative weights
• In Kruskal’s algorithm, adding an edge without checking for cycles — forgetting to apply the Union-Find check means the output may not be a tree
• In topological sort, applying it to a graph that contains a cycle — topological ordering is only defined for directed acyclic graphs (DAGs)
• Mixing up BFS and DFS traversal orders when doing them by hand — BFS uses a queue (FIFO) while DFS uses a stack (LIFO) or recursion