Graph Representation

Adjacency List

What is an Adjacency List?

An adjacency list represents a graph as a collection of per-vertex neighbor lists: one entry per vertex, holding only the vertices it's actually connected to. Instead of a full V×V grid of mostly zeros, each vertex stores exactly as many entries as it has edges — nothing more.

How Does It Work?

Adding an edge (u, v) appends v to u's list. For an undirected graph, u is also appended to v's list, since the edge goes both ways; for a directed graph, only u's list gets the new entry. Checking whether an edge exists means scanning through one vertex's list looking for the target — proportional to that vertex's degree (its number of neighbors), not the whole graph.

This is the mirror image of an adjacency matrix's tradeoffs. A matrix spends O(V²) space no matter what, in exchange for O(1) edge-existence checks. A list spends space proportional to the actual number of edges — O(V + E) — but checking a specific edge now costs O(degree) instead of O(1). For the vast majority of real-world graphs, which are sparse (E is much smaller than V²), that tradeoff strongly favors the list.

The same triangle graph — every vertex's list holds only its actual neighbors
ABC
A→
BC
B→
AC
C→
AB

Building the List

  1. Create an empty list (or map) for every vertex
  2. For every edge (u, v) with weight w: append (v, w) to u's list
  3. If the graph is undirected, also append (u, w) to v's list — the same edge is recorded from both directions
  4. To check if an edge exists between two vertices, scan the source vertex's list for the target

Complexity

  • Space Complexity: O(V + E) — proportional to the actual number of vertices and edges, not V².
  • Check if edge (u, v) exists: O(degree(u)) — scan u's list, which is only as long as u's actual neighbor count.
  • Iterate over a vertex's neighbors: O(degree(u)) — the list already holds exactly the relevant entries.
  • Add an edge: O(1) — appending to a list.

Time Complexity Analysis

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Adjacency lists are the default choice for most graph algorithms — BFS, DFS, Dijkstra's algorithm, and topological sort all need to repeatedly ask "what are this vertex's neighbors?", which a list answers by directly returning exactly the relevant entries, without wasting time scanning past vertices that aren't connected at all.

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Adjacency List

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Neighbor entryMost recently added edgeSelected (click any entry)

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Adjacency List Implementation

class GraphList {
  constructor() {
    this.list = new Map(); // vertex -> [{ to, weight }, ...]
  }

  addVertex(v) {
    if (!this.list.has(v)) this.list.set(v, []);
  }

  addEdge(from, to, weight = 1, directed = false) {
    this.list.get(from).push({ to, weight });
    if (!directed) this.list.get(to).push({ to: from, weight }); // mirror the edge both ways
  }

  hasEdge(from, to) {
    return this.list.get(from).some((entry) => entry.to === to); // O(degree(from))
  }

  neighbors(v) {
    return this.list.get(v).map((entry) => entry.to); // exactly the relevant entries
  }
}

// Usage example
const g = new GraphList();
["A", "B", "C", "D"].forEach((v) => g.addVertex(v));
g.addEdge("A", "B");
g.addEdge("A", "C");
g.hasEdge("A", "B"); // true
g.neighbors("A");    // ["B", "C"]

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