Clique graphs of time graphs

Abstract A simple, finite graph G is called a time graph (equivalently, an indifference graph ) if there is an injective real function f on the vertices v ( G ) such that v i v j ∈ e ( G ) for v i ≠ v j if and only if | f ( v i ) − f ( v j )| ≤ 1. A clique of a graph G is a maximal complete subgraph of G . The clique graph K(G) of a graph G is the intersection graph of the cliques of G . It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K 2 ( G ), and inductively, denote K ( K m −1 ( G )) by K m ( G ). Define the index indx( G ) of a connected time graph G as the smallest integer n such that K n ( G ) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.