Problems tagged with "connected components"

An undirected graph with 20 nodes and 30 edges has three connected components: \(A\), \(B\), and \(C\). Connected component \(A\) has 7 nodes. What is the smallest number of edges it can have?

Let \(G = (V, E)\) be an undirected graph, and suppose that \(u, v, \) and \(w\) are three nodes in \(V\). Suppose that \(u\) and \(v\) are in the same connected component, but \(u\) and \(w\) are not in the same connected component. True or False: it is possible that \(v\) and \(w\) are in the same connected component.

Let \(C\) be a connected component in an undirected graph \(G\). Suppose \(u_1\) is a node in \(C\) and let \(u_2\) be a node that is reachable from \(u_1\). True or False: \(u_2\) must also be in the same connected component, \(C\).

Suppose \(G = (V, E)\) is an undirected graph. Let \(k\) be the number of connected components in \(G\). True or False: it is possible that \(k > |E|\).

Suppose an undirected graph \(G = (V,E)\) has two connected components, \(C_1\) and \(C_2\). Suppose that \(|C_1| = 4\) and \(|C_2| = 3\). What is the largest that \(|E|\) can possibly be?