How Graph Labs works

A two-minute walkthrough to get to know the studio with a ready-made graph:

1. 1.In the Build tab, click Weighted network. The graph is loaded and framed automatically.
2. 2.Go to Run and choose Dijkstra's algorithm, under Shortest paths.
3. 3.In Root / origin, select A; in Target vertex, select F.
4. 4.Click Run Dijkstra and use Next step to watch each vertex get closed and each tense edge get relaxed in the dist and pred table.
5. 5.At the last step, the shortest path A → C → F, with weight 11, turns purple, and the Conclusions card sums up the final distances.

Only one tool is active at a time, highlighted in the top bar. The cursor changes shape to show which one is in use, and a hint next to the bar explains what to do.

• Select and move

  The default tool. Click a vertex or an edge to select it, drag vertices to reposition them and drag the background to move the view.

• Add vertex

  Each click on an empty spot creates a vertex there. Labels follow the sequence A, B, C, ..., Z, A1, B1, ..., always skipping the ones already in use.

• Connect vertices

  Click the source vertex and then the target. Between the two clicks, a dashed line follows the cursor. Esc cancels.

• Remove element

  Click a vertex or an edge to delete it. Removing a vertex also removes every edge attached to it.

• Undo

  Reverts the last change to the graph. Keeps up to 60 changes.

• Redo

  Reapplies a change that was undone.

• Clear graph

  Deletes every vertex and edge. It can be reverted with Undo.

• New edges are undirected

  Edges created on the canvas start undirected (default).

• New edges are directed

  Edges created on the canvas start with an arrow, from source to target.

• Layout suggestion

  When on, each new edge triggers a light adjustment of the drawing. The preference is saved.

• Rearrange now

  Applies the layout adjustment right away, just once.

Drag the background to move the view and use the mouse wheel (or a pinch gesture on a trackpad or phone) to zoom in and out, always centered on the point under the cursor. Zoom ranges from 30% to 260%. The buttons in the bottom-right corner offer the same control:

• Zoom in

  Increases the zoom by 25%, keeping the center.

• Zoom out

  Decreases the zoom by 20%, keeping the center.

• Fit graph

  Adjusts zoom and position so the whole graph fits on screen.

When a sample graph is loaded, it is framed automatically. Parallel edges between the same pair of vertices, such as A → B and B → A, are drawn curved so they do not overlap.

Once an algorithm runs, every vertex and edge gets a state at each step. The legend in the bottom-left corner of the canvas sums up what the colors mean:

• Unexplored

  Initial state: the algorithm has not reached the element yet.

• Marked

  Reached but not processed yet: it is in the queue, the stack or the frontier.

• Under analysis

  Element examined in the current step. Vertices under analysis pulse to draw attention.

• Explored / in solution

  Processing finished or element accepted into the solution (tree, order, matching).

• Discarded

  Rejected by the algorithm, such as an edge that would form a cycle. Discarded edges are dashed.

• Path

  Result highlighted at the end: shortest path, augmenting path or Eulerian trail.

• Dashed ring in the main color

  Starting vertex. The label above it reads ROOT or, in flow algorithms, SOURCE.

• Purple dashed ring

  Ending vertex: TARGET, or SINK in flow algorithms.

• 1/6

  Badge under the vertex

  Value of the vertex at the current step: d/f in depth-first search, level in breadth-first search, dist in Dijkstra, color in coloring, s and t in flow.

• 3/5

  Edge label

  Shows the weight. During a run it may give way to another value, such as flow/capacity in flow algorithms or the traversal order in Fleury's algorithm.

• Colored outline

  Groups vertices of the same set: strongly connected components in Kosaraju, trees of the forest in Kruskal, color classes in coloring.

The samples reproduce examples used in class, each one designed to highlight the behavior of specific algorithms. Loading a sample replaces the current graph (you can undo it), clears the chosen root and target and frames the drawing.

• Weighted network

  Undirected weighted graph, with weight w(e) > 0 on every edge: the base for MSTs (Prim and Kruskal) and for Dijkstra.

  PrimKruskalDijkstra

• Directed graph with cycles

  Directed graph with three strongly connected components, for Kosaraju's algorithm.

  KosarajuDFS

• Flow network

  Flow network: a directed graph with capacity u(e) on every edge, from the source s = S to the sink t = T.

  Ford-Fulkerson

• Negative weights

  Directed graph with negative-weight edges and no negative-weight cycle, for Bellman-Ford and Floyd-Warshall.

  Bellman-FordFloyd-Warshall

• Simple graph

  Simple undirected graph with no relevant weights: a good fit for breadth-first and depth-first search.

  BFSDFS

• Eulerian graph

  Example 1 from the Eulerian graphs lecture: every vertex has even degree, so an Eulerian circuit exists.

  Fleury

• Semi-Eulerian graph

  Example 2 from the lecture: exactly two vertices of odd degree (5 and 6), so an open Eulerian trail exists.

  Fleury

• Network with a bottleneck

  The network from the Edmonds-Karp lecture: two edges of capacity 100 joined by one of capacity 1, which exposes the weakness of choosing paths arbitrarily.

  Ford-FulkersonEdmonds-KarpDinic

• Activity precedence

  Acyclic graph from the topological sorting lecture: building a bookshelf, from buying the boards to moving it.

  KahnTopological sort (DFS)

• Matching with blossoms

  General graph with two odd-length cycles: it requires the blossom contraction of Edmonds' algorithm.

  Edmonds

• Vertex coloring

  Graph with χ(G) = 3 in which alphabetical order makes the greedy method use 4 colors, while Welsh-Powell finds 3.

  Greedy coloringWelsh-Powell

• Welsh-Powell counterexample

  Bipartite graph, so χ(G) = 2, on which Welsh-Powell still uses 3 colors. It is the counterexample from the coloring lecture.

  Welsh-PowellGreedy coloring

The Vertices card lists every vertex in alphabetical order, with the total count in the header. The New button creates a vertex on the canvas without switching tools; then just drag it where you want it.

• Rename: edit the label directly in the text field, up to 6 characters. The alphabetical order of the labels is the default visit order of every algorithm.
• Select: click the circle with the initials or the text field to highlight the vertex on the canvas.
• Remove: the trash icon deletes the vertex and every edge incident to it.

The form at the top of the card creates edges precisely, which helps with large graphs or when copying an exercise: choose the From and To vertices, enter the Weight and the Type (undirected or directed) and click Add edge. The header shows the total number of edges and how many there are of each type.

Each edge in the list can be edited without being recreated:

• the numeric field changes the weight, and clearing it leaves the edge unweighted;
• the badge toggles the direction with one click: undirected / directed
• clicking the labels selects the edge on the canvas;
• the trash icon removes the edge.

The Algorithm card groups the methods by topic, in course order. Each option shows the name, the complexity and a summary of the strategy. The selected algorithm is highlighted and defines the content of the Parameters card right below.

The card header repeats the name and complexity of the method, followed by the graph requirements as badges. Vertex fields only appear when the algorithm uses them:

When a required field has not been chosen yet, the studio uses the first vertex in alphabetical order (and, for the sink, the first one different from the source). The chosen vertices get a dashed ring on the canvas with the label ROOT, TARGET, SOURCE or SINK.

Many algorithms need to decide which neighbor to examine first. By default the decision follows the alphabetical order of the labels, which is the convention used in class. When the exercise asks for another order, build it in Visit sequence:

• click the vertices in the desired order to add them to the sequence;
• the first vertex chosen becomes the root when none is set above (in flow algorithms, it is the first neighbor tried in the search);
• click a vertex in the sequence to take it out;
• vertices left out follow in alphabetical order, after the chosen ones;
• the Default button goes back to alphabetical order.

Requirements are checked on every change to the graph or the parameters. If everything is fine, a green confirmation appears; otherwise, each problem is listed in red with the suggested fix, and the Run button is disabled.

The header shows the algorithm and the current position (for example, Step 4 of 23), with a progress bar right below. The controls are:

• First step

  Goes back to the initial state.

• Previous step

  Goes back one iteration.

• Play / pause

  Moves forward on its own at the chosen pace. At the end, it restarts from the first step.

• Next step

  Moves forward one iteration.

• Last step

  Jumps to the final result.

• Clear

  Discards the run and restores the original canvas colors.

The slider jumps straight to any step. Any manual navigation pauses automatic playback. The available speeds are:

The main card shows the title of the decision taken (for example, "Tense edge (A, C): relaxed"), the reasoning with the values involved and, when it makes sense, metrics such as the visit order, the current iteration or the flow value. The algorithm’s auxiliary structures appear below it.

The tables reproduce the ones on the board: dist and pred, discovery and finish times, Floyd-Warshall matrices, flow and residual capacities, among others. Colored rows show the role of each entry at the current step:

At the last step the green Conclusions card appears and interprets the result: final distances and the recovered path, total weight of the spanning tree, maximum flow value and the matching cut, components found, topological order, number of colors used. It is the summary to compare with your answer.