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Let T be an edge weighted tree and d min , d max be two non-negative real numbers where d min d max : The pairwise compatibility graph (PCG) of T for d min , d max is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval ½d min , d max : A graph G is a PCG if there exist an edge weighted tree T and suitable d min , d max such that G is a PCG of T. The class of pairwise compatibility graphs was introduced to model evolutionary relationships among a set of species. Since not all graphs are PCGs, researchers become interested in recognizing and characterizing PCGs. In this paper, we review the results regarding PCGs and some of its variants.

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Let [Formula: see text] be a set of [Formula: see text] customers and [Formula: see text] be a set of [Formula: see text] facilities. An [Formula: see text]-gather clustering of [Formula: see text] is a partition of the customers in clusters such that each cluster contains at least [Formula: see text] customers. The [Formula: see text]-gather clustering problem asks to find an [Formula: see text]-gather clustering which minimizes the maximum distance between a pair of customers in a cluster. An [Formula: see text]-gathering of [Formula: see text] to [Formula: see text] is an assignment of each customer [Formula: see text] to a facility [Formula: see text] such that each facility has zero or at least [Formula: see text] customers. The [Formula: see text]-gathering problem asks to find an [Formula: see text]-gathering that minimizes the maximum distance between a customer and his/her facility. In this work, we consider the [Formula: see text]-gather clustering and [Formula: see text]-gathering problems when the customers and the facilities are lying on a “star”. We show that the [Formula: see text]-gather clustering problem and the [Formula: see text]-gathering problem with customers and facilities on a star with [Formula: see text] rays can be solved in [Formula: see text] and [Formula: see text] time, respectively. Furthermore, we prove the hardness of a variant of the [Formula: see text]-gathering problem, called the min-max-sum [Formula: see text]-gathering problem, even when the customers and the facilities are on a star. We also study the [Formula: see text]-gathering problem when the customers and the facilities are on a line, and each customer location is uncertain. We show that the [Formula: see text]-gathering problem can be solved in [Formula: see text] and [Formula: see text] time when the customers and the facilities are on a line, and the customer locations are given by piecewise uniform functions of at most [Formula: see text] pieces and “well-separated” uniform distribution functions, respectively.