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Computer Science · Algorithms and Search

Graph Representation and Adjacency Intuition

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Breadth-First Search and Layered FrontiersUse adjacency with a queue frontier

Key takeaway

Adjacency is the local neighborhood information available from the current node.
The first frontier widens or narrows because different graph starts have different neighbor sets.
Traversal rules reuse the same graph structure but change the order in which the frontier is handled.

Common misconception

A graph algorithm starts by applying a queue or stack rule before local structure matters.

The honest first move is local: read the direct neighbors of the current node.

Turn direct neighbors into the next frontier

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

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

Use one cyclic graph to see which nodes are waiting, which are finished, and why repeat edges are skipped instead of reopening loops.

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Reference and support

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Equation snapshotNeighbor set · First frontierShow 2 equations

Neighbor set
$N (v) = {u : (v, u) \in E}$
Lists the nodes you can reach from v by one edge.
First frontier
$F_{1} = N (s)$
The first frontier is exactly the start node's neighbor set.

Why it behaves this way

Explanation

A graph should first feel like a map of direct connections, not a picture you have to understand all at once. This bench keeps one labeled graph, one active frontier, and one adjacency readout together so you can read the current neighborhood before the search language gets more formal.

Adjacency is the first useful question: which nodes can the current node reach by one edge right now? Once that local neighborhood is clear, breadth-first and depth-first search become two ways of organizing the same next-step options.

Key ideas

01The first useful graph reading is local adjacency: which nodes are one edge away from the current node.

02The frontier is the set of discovered next candidates that are waiting to be explored.

03Changing the start node changes the first neighborhood and the first frontier, even when the graph itself stays the same.

Worked examples

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Frozen walkthrough

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Frozen walkthrough

Read the first neighborhood from the bench before you name BFS or DFS.

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Frozen valuesUsing frozen parameters

On the layered campus graph, what does starting at A tell you immediately?

G

Graph layout

Layered campus

s

Start node

F_{1}

Traversal mode

B, C

1. Read the direct neighbors

Node A touches B and C directly, so those two nodes are the first adjacent candidates.

2. Turn those neighbors into the first frontier

After the first expansion, the claimed next-step frontier is [B, C] rather than the whole graph.

3. Connect that local picture to later search rules

Breadth-first and depth-first search will organize that frontier differently, but they still start from the same local adjacency picture.

First-neighborhood summary

From A, the first frontier is B and C.

Adjacency is the local structure that every later search rule has to respect.

Quick test

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Featured setups

Layered campus from A

Open a graph where A has only a small first neighborhood, so the first adjacency read is easy to follow.

Hub and detours from C

Start from the hub so the first step immediately reveals several next nodes.

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Starter track

Step 3 of 6

Algorithms and Search Foundations

Graph Representation and Adjacency Intuition appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Binary Search / Halving the Search Space

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