Questions: (a) Find the connected components of each graph. (i) G=(V, E). V=a, b, c, d, e. E=∅ (ii) G=(V, E). V=a, b, c, d, e, f. E=c, f,a, b,d, a,e, c,b, f

Solution Steps

To find the connected components of a graph, we need to identify subsets of vertices such that there is a path between any two vertices within the same subset, and no path exists between vertices in different subsets. For each graph, we can use a depth-first search (DFS) or breadth-first search (BFS) to explore and group connected vertices.

(i) Since the edge set \( E \) is empty, each vertex is its own connected component.

(ii) We can use a graph traversal algorithm to explore the graph and identify connected components by visiting all vertices reachable from each unvisited vertex.

For the graph \( G = (V, E) \) where \( V = \{a, b, c, d, e\} \) and \( E = \emptyset \), there are no edges connecting any vertices. Therefore, each vertex is isolated and forms its own connected component. The connected components are: \[ \{a\}, \{b\}, \{c\}, \{d\}, \{e\} \]

For the graph \( G = (V, E) \) where \( V = \{a, b, c, d, e, f\} \) and \( E = \{ \{c, f\}, \{a, b\}, \{d, a\}, \{e, c\}, \{b, f\} \} \), we can identify the connected components by examining the edges. The edges connect the vertices as follows:

• \( c \) is connected to \( f \) and \( e \)
• \( a \) is connected to \( b \) and \( d \)
• \( b \) is also connected to \( f \)

By traversing the graph, we find that all vertices are interconnected, forming a single connected component: \[ \{a, b, c, d, e, f\} \]

Final Answer