Abstract
Combinatorics 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.
Highlights
Let [n] := {1, 2, . . . , n} and let k ≤ n be a positive integer
This is in part for their applications in Network Design–where connectivity and diameter (i) are of importance
In this paper we study the connectedness and diameter of a “geometric” version of the generalized Johnson graph
Summary
Let [n] := {1, 2, . . . , n} and let k ≤ n be a positive integer. A k-subset of a set is a subset of k elements. The Johnson graph J(n, k) is the graph whose vertex set consists of all k-subsets of [n], two of which are adjacent if their intersection has size k − 1. The generalized Johnson graph GJ(n, k, l) is the graph whose vertex set consists of all k-subsets of [n], two of which are adjacent if they have exactly l elements in common. Johnson graphs have been widely studied in the literature This is in part for their applications in Network Design–where connectivity and diameter (i) are of importance. In this paper we study the connectedness and diameter of a “geometric” version of the generalized Johnson graph. The generalized island Johnson graph IJ(P, k, l) is the graph whose vertex set consists of all k-islands of P , two of which are adjacent if their intersection has exactly l elements. A preliminary version of this paper appeared in Bautista-Santiago et al (2010)
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