Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves { 1 {combining short stroke overlay}ñ} obtained from n continuous functions f_i:[0,1]→R(1≤i≤n). We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement Ḡ is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ≤k permutation diagrams if and only if the partial order dimension of G ̄ is ≤k+1. Computational complexity results are obtained for recognizing such graphs.