Neighbor set
The adjacent nodes of v are the nodes joined to it by one edge.
Graph layout 0 Start node 0
Concept module
Graph Representation and Adjacency Intuition
Keep one live graph, one local neighborhood, and one frontier cue visible together so graph structure feels readable before traversal rules get formal.
The simulation shows one labeled graph with the current node, the frontier nodes, and the visited nodes colored differently so local neighborhoods stay visible. A readout card reports the traversal mode, the current node, the frontier size, and the target, while a cue panel shows the frontier order and the current neighbor list. Breadth-first search is running on the layered campus graph from A toward H. The start node is waiting on the frontier. The frontier currently holds 1 node, and 0 nodes have already been visited.
Interactive lab
Graph traversal bench
Keep one live graph, the current frontier, and the visited state visible together so breadth-first and depth-first search read like different process choices on the same structure.
Controls
Swap among a few bounded graph scenes without leaving the shared bench.
Choose where the traversal begins.
Mark the node the search is trying to reach.
Compare queue-like breadth-first expansion with stack-like depth-first branching.
More tools
Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.
Show
More tools
Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.
More presets
Presets
Time
0.00 s / 20.3 sLivePause to inspect a specific moment, then step or scrub through it.0.00 s20.3 s
Predict -> manipulate -> observe
Keep the active prompt next to the controls so each change has an immediate visible consequence.
ObservationPrompt 1 of 2
The first honest reading is local: which nodes touch the current node directly right now.
Graphs
Switch graph views without breaking the live stage and time link.
Visited nodes versus frontier size
Watch how many nodes have already been expanded while the waiting frontier grows or shrinks.
step: -1.62 to 21.9nodes: -0.64 to 8.64
visited nodesfrontier size
Equation map
See each variable before you move it.
Select a symbol to highlight the matching control and the graph or overlay it most directly changes.
Graph layout
0Switches among the bounded graph scenes on the same traversal bench.
Graph: Visited nodes versus frontier sizeGraph: New discoveries versus repeat skipsOverlay: Adjacency cueOverlay: Visited cue
Equations in play
Choose an equation to sync the active symbol, control highlight, and related graph mapping.
More tools
Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.
Hide
More tools
Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.
What to notice
Keep the graph and one traversal graph visible together.
ObservationPrompt 1 of 2
Graph: Visited nodes versus frontier size
The first honest reading is local: which nodes touch the current node directly right now.
Control: GraphControl: Start nodeGraph: Visited nodes versus frontier sizeOverlay: Adjacency cueOverlay: Frontier cueEquationEquation
Guided overlays
Focus one overlay at a time to see what it represents and what to notice in the live motion.
3 visible
Overlay focus
Adjacency cue
Keep the current node's neighbor list visible.
What to notice
- The local neighborhood is the first honest input to every search step.
Why it matters
It keeps graph reading local and visual instead of turning the diagram into a vague picture.
Control: GraphControl: Start nodeGraph: New discoveries versus repeat skipsEquationEquation
Breadth-first search is running on the layered campus graph from A toward H. The start node is waiting on the frontier. The frontier currently holds 1 node, and 0 nodes have already been visited.
Equation detailsDeeper interpretation, notes, and worked variable context.
Degree
The degree counts how many direct neighbors a node has.
Graph layout 0
First frontier
The first frontier is the neighborhood of the chosen start node.
Start node 0 Target node 7 Traversal mode 0
Progress
Not startedMastery: NewLocal-firstStart exploring and Open Model Lab will keep this concept's progress on this browser first. No finished quick test, solved challenge, or completion mark is saved yet.
Let the live model runChange one real controlOpen What to notice
Try this setup
Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.
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Current bench
Layered campus BFS preset
This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.
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Stable links
Starter track
Step 3 of 60 / 6 completeAlgorithms and Search Foundations
Earlier steps still set up Graph Representation and Adjacency Intuition.
1. Sorting and Algorithmic Trade-offs2. Binary Search / Halving the Search Space3. Graph Representation and Adjacency Intuition4. Breadth-First Search and Layered Frontiers+2 more steps
Previous step: Binary Search / Halving the Search Space.
Short explanation
What the system is doing
A graph should feel like a live neighborhood map, not an abstract list of circles and lines. This bench keeps one labeled graph, one active frontier, and one adjacency readout together so local connections stay readable before the search language gets more formal.
Adjacency is the first honest question: which nodes can the current node touch directly right now? Once that local neighborhood is visible, breadth-first and depth-first search become different ways of organizing the same next-step options.
Key ideas
01A graph is built from nodes and edges, but the useful reading is local adjacency.
02The frontier is just the set of adjacent next candidates that have already been claimed but not fully explored.
03Changing the start node changes the first neighborhood even when the graph itself stays the same.
Worked example
Read the full frozen walkthrough.
Frozen walkthrough
Read one neighborhood directly from the bench before you talk about algorithms.
Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.
View plansFrozen valuesUsing frozen parameters
On the layered campus graph, what does the start node A make visible immediately?
Graph layout
Layered campus
Start node
A
First frontier
B, C
1. Read only the direct neighbors first
Node A touches B and C directly, so those two nodes are the first adjacent candidates.
2. Turn that neighborhood into a frontier
After the first expansion, the claimed next-step frontier is [B, C] rather than the whole graph.
3. Name what stays the same later
Breadth-first and depth-first search will organize that frontier differently, but they still start from the same local adjacency picture.
First neighborhood read
From A, the first frontier is B and C.
Adjacency is the local structure that every later search rule has to respect.
Common misconception
Reading a graph means looking at the whole diagram at once and guessing a path globally.
The honest first move is local: read the neighbors of the current node.
Search grows from one neighborhood to the next, not from a magical full-map scan.
Mini challenge
Move the start node until the first neighborhood clearly widens.
Prediction prompt
Decide whether you need a hub node or an edge node before you move the slider.
Check your reasoning
Start from a hub node if you want the first frontier to widen immediately.
A hub has more adjacent neighbors, so the frontier fans out sooner even though the graph is the same object.
Quick test
Reasoning
Question 1 of 2
Answer from the visible neighborhood, not from memorized graph jargon.
What does adjacency mean on this graph bench?
Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.
Accessible description
The simulation shows one labeled graph with the current node, the frontier nodes, and the visited nodes colored differently so local neighborhoods stay visible.
A readout card reports the traversal mode, the current node, the frontier size, and the target, while a cue panel shows the frontier order and the current neighbor list.
Graph summary
One graph tracks visited nodes against frontier size, a second tracks the current depth against the deepest claimed depth, and a third compares new discoveries with repeat skips.
The graph hover and the step-through-time rail stay tied to the same live traversal bench.
Turn local neighborhoods into a live search frontier
Keep moving through the concept map.
These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.
Turn local adjacency into a layered searchAlgorithms and Search
Breadth-First Search and Layered Frontiers
Keep the queue frontier, visited count, and graph layers visible together so breadth-first search reads as a layered process instead of a procedure list.
Compare the same graph under a branch-first frontierAlgorithms and Search
Depth-First Search and Backtracking Paths
Keep the stack frontier, current depth, and branch order visible together so depth-first search feels like disciplined backtracking instead of random wandering.
Next in Cycle control and visited stateAlgorithms and Search
Frontier and Visited State on Graphs
Keep repeat skips, waiting frontier nodes, and already-expanded nodes visible together so cycle handling feels like honest bookkeeping on one graph bench.