Problem 1: A k-tree is a type of chordal graph given by the following recursive definition. (1) Kk+1 is a k-tree. (2) Let G be a k-tree. Then G + v is a k-tree if the neighbors of v in G form a k-clique. Prove that every tree with at least 2 nodes is a 1-tree. Problem 2: Prove that every cycle is a subgraph of a 2-tree. Problem 3: Prove that being a subgraph of a k-tree is a hereditary property.
