1.1. Various Operations on Graphs

Try these first

1. 1.

   Define a subgraph.

   Hint

   Think about the relationship between parts and wholes.

2. 2.

   What is the difference between a subgraph and a proper subgraph?

   Hint

   Consider the definitions carefully.

3. 3.

   What defines a proper subgraph?

   • It must equal the original graph
   • It must have fewer vertices or edges
   • It must be disconnected

   Hint

   Think about the specific criteria.

4. 4.

   True or False: An induced subgraph can have all edges of the original graph.

   • True
   • False

   Hint

   Remember the definition of induced subgraphs.

5. 5.

   Given a graph with vertices A, B, C, and D, where edges AB, AC, and CD exist, identify all possible subgraphs. Is there a proper subgraph?

   Hint

   Consider combinations of vertices and their edges.

6. 6.

   Discuss whether graphs can be isomorphic without having the same number of edges. Provide the reasoning.

   Hint

   Think about the structural properties of graphs.