Recognizing edge clique graphs among interval graphs and probe interval graphs

The edge clique graph of a graph HH is the one having the edge set of HH as vertex set, two vertices being adjacent if and only if the corresponding edges belong to a common complete subgraph of HH. We characterize the graph classes {edge clique graphs} ∩∩ {interval graphs} as well as {edge clique graphs} ∩∩ {probe interval graphs}, which leads to polynomial time recognition algorithms for them. This work generalizes corresponding results in [M.R. Cerioli, J.L. Szwarcfiter, Edge clique graphs and some classes of chordal graphs, Discrete Math. 242 (2002) 31–39].