String graphs (intersection graphs of curves in the plane) were originally studied in connection with RC-circuits. The family of string graphs is closed in the induced minor order, and so it is reasonable to study critical nonstring graphs (nonstring graphs such that all of their proper induced minors are string graphs). The question of whether there are infinitely many nonisomorphic critical nonstring graphs has been an open problem for some time. The main result of this paper settles this question. In a later paper of this series we show that recognizing string graphs is NP-hard. © 1991.
Kratochvíl, J. (1991). String graphs. I. The number of critical nonstring graphs is infinite. Journal of Combinatorial Theory, Series B, 52(1), 53–66. https://doi.org/10.1016/0095-8956(91)90090-7