Linear edge bound for K_{t,t}-free curve visibility graphs

Establish whether every n-vertex K_{t,t}-free curve visibility graph has O_t(n) edges, where a curve visibility graph is formed by points on a Jordan curve with adjacency defined by mutual straight-line visibility through the curve’s interior.

Background

The paper explains that polygon visibility graphs are not hereditary: an induced subgraph of a polygon visibility graph need not itself be the visibility graph of another polygon. Induced subgraphs of polygon visibility graphs are equivalent to curve visibility graphs, defined using points on a Jordan curve and straight-line visibility.

The cited open problem asks whether the quasi-linear behavior proved in the paper for more general curve pseudo-visibility graphs can be improved to a linear bound for ordinary curve visibility graphs. The paper does not resolve this question; instead, it gives an Ω(nα(n)) lower bound for the more general class of curve pseudo-visibility graphs and an O_t(n log n) upper bound for that class.