Which graph representation uses space proportional to V + E, where V is the number of vertices and E is the number of edges?

Which graph representation uses space proportional to V + E, where V is the number of vertices and E is the number of edges?

• A. Adjacency matrix
• B. Edge list
• C. Incidence matrix
• D. Adjacency list

Answer: (D)

Storing a graph efficiently depends on how you access its neighbors. An adjacency list keeps a separate list for each vertex containing the vertices it’s connected to. You pay a tiny header for every vertex, plus one entry for each edge incident to that vertex. Across the whole graph, that sums to about V plus E entries, so the total space grows proportionally with V + E. This makes it especially space-efficient for sparse graphs where E is much smaller than V^2, and it also makes traversing neighbors quick. In contrast, an adjacency matrix reserves space for every possible pair of vertices, which is V^2. That can waste a lot of space when the graph is sparse. An incidence matrix uses a grid with V rows and E columns, leading to V × E space, which can be very large as the graph grows. An edge list stores each edge as a pair of vertices, needing space proportional to E alone (plus any extra structure to map vertices), which doesn’t scale with V in the same additive way as V + E and can be less convenient for frequent neighbor lookups.

Storing a graph efficiently depends on how you access its neighbors. An adjacency list keeps a separate list for each vertex containing the vertices it’s connected to. You pay a tiny header for every vertex, plus one entry for each edge incident to that vertex. Across the whole graph, that sums to about V plus E entries, so the total space grows proportionally with V + E. This makes it especially space-efficient for sparse graphs where E is much smaller than V^2, and it also makes traversing neighbors quick.

In contrast, an adjacency matrix reserves space for every possible pair of vertices, which is V^2. That can waste a lot of space when the graph is sparse. An incidence matrix uses a grid with V rows and E columns, leading to V × E space, which can be very large as the graph grows. An edge list stores each edge as a pair of vertices, needing space proportional to E alone (plus any extra structure to map vertices), which doesn’t scale with V in the same additive way as V + E and can be less convenient for frequent neighbor lookups.