Given a set P of n points and an integer k, we wish to place k facilities on points in P so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem, and the set of such k points is called a k-dispersion of P. Note that the 2-dispersion problem corresponds to the computation of the diameter of P. Thus, the k-dispersion problem is a natural generalization of the diameter problem. In this paper, we consider the case of k=3, which is the 3-dispersion problem, when P is in convex position. We present an O(n2)-time algorithm to compute a 3-dispersion of P.

Given a set P of n points on which facilities can be placed and an integer k, we want to place k facilities on some points so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem. In this paper, we consider the 3-dispersion problem when P is a set of points on a plane (2-dimensional space). Note that the 2-dispersion problem corresponds to the diameter problem. We give an O(n) time algorithm to solve the 3-dispersion problem in the L∞ metric, and an O(n) time algorithm to solve the 3-dispersion problem in the L1 metric. Also, we give an O(n2 log n) time algorithm to solve the 3-dispersion problem in the L2 metric.

In this study, we address a problem pertaining to the induced matching enumeration. An edge set M is an induced matching of a graph G=(V,E). The enumeration of matchings has been widely studied in literature; however, there few studies on induced matching. A straightforward algorithm takes O(Δ2) time for each solution that is coming from the time to generate a subproblem, where Δ is the maximum degree in an input graph. To generate a subproblem, an algorithm picks up an edge e and generates two graphs, the one is obtained by removing e from G, the other is obtained by removing e, adjacent edge to e, and edges adjacent to adjacent edge of e. Since this operation needs O(Δ2) time, a straightforward algorithm enumerates all induced matchings in O(Δ2) time per solution. We investigated local structures that enable us to generate subproblems within a short time and proved that the time complexity will be O(1) if the input graph is C4-free. A graph is C4-free if and only if none of its subgraphs have a cycle of length four.

We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n2|ε(S)|)-time, where ε(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n2)-delay, while the known algorithm enumerates in O(n2 log n)-delay, where a surroundingpolygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.

By the motivation to discover patterns in massive structured data in the form of graphs and trees, we study a special case of the k-subtree enumeration problem with a tree of n nodes as an input graph, which is originally introduced by (Ferreira, Grossi, and Rizzi, ESA'11, 275-286, 2011) for general graphs. Based on reverse search technique (Avis and Fukuda, Discrete Appl. Math., vol.65, pp.21-46, 1996), we present the first constant delay enumeration algorithm that lists all k-subtrees of an input rooted tree in O(1) worst-case time per subtree. This result improves on the straightforward application of Ferreira et al.'s algorithm with O(k) amortized time per subtree when an input is restricted to tree. Finally, we discuss an application of our algorithm to a modification of the graph motif problem for trees.

A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.