Let S be a bicolored set of n points on the plane. A subset I ⊆ S is called an island of S, if I is the intersection of S and a convex set C. In this paper we give an O(n 3 )-time algorithm to find a monochromatic island of maximum cardinality. Our approach also optimizes other parameters and gives an approximation to the class cover problem.

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.

In this paper we study the extremal type problem arising from the question: What is the maximum number E T (S) of edges that a geometric graph G on a Author's personal copyGraphs and Combinatorics planar point set S can have such that it does not contain empty triangles? We prove: n 2 − O(n log n) ≤ E T (n) ≤ n 2 − n − 2 + n 8 .

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Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P ∩ C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.