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.