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.

The problem of enumerating connected induced subgraphs of a given graph is classical and studied well. It is known that connected induced subgraphs can be enumerated in constant time for each subgraph. In this paper, we focus on highly connected induced subgraphs. The most major concept of connectivity on graphs is vertex connectivity. For vertex connectivity, some enumeration problem settings and enumeration algorithms have been proposed, such as k-vertex connected spanning subgraphs. In this paper, we focus on another major concept of graph connectivity, edge-connectivity. This is motivated by the problem of finding evacuation routes in road networks. In evacuation routes, edge-connectivity is important, since highly edge-connected subgraphs ensure multiple routes between two vertices. In this paper, we consider the problem of enumerating 2-edge-connected induced subgraphs of a given graph. We present an algorithm that enumerates 2-edge-connected induced subgraphs of an input graph G with n vertices and m edges. Our algorithm enumerates all the 2-edge-connected induced subgraphs in O(n3m|SG|) time, where SG is the set of the 2-edge-connected induced subgraphs of G. Moreover, by slightly modifying the algorithm, we have a O(n3m)-delay enumeration algorithm for 2-edge-connected induced subgraphs.

In this paper, we present an algorithm that counts the number of empty quadrilaterals whose corners are chosen from a given set S of n points in general position. Our algorithm can separately count the number of convex or non-convex empty quadrilaterals in O(T) time, where T denotes the number of empty triangles in S. Note that T varies from Ω(n2) and O(n3) and the expected value of T is known to be Θ(n2) when the n points in S are chosen uniformly and independently at random from a convex and bounded body in the plane. We also show how to enumerate all convex and/or non-convex empty quadrilaterals in S in time proportional to the number of reported quadrilaterals, after O(T)-time preprocessing.

We investigate enumeration of distinct flat-foldable crease patterns under the following assumptions: positive integer n is given; every pattern is composed of n lines incident to the center of a sheet of paper; every angle between adjacent lines is equal to 2π/n; every line is assigned one of “mountain,” “valley,” and “flat (or consequently unfolded)”; crease patterns are considered to be equivalent if they are equal up to rotation and reflection. In this natural problem, we can use two well-known theorems for flat-foldability: the Kawasaki Theorem and the Maekawa Theorem in computational origami. Unfortunately, however, they are not enough to characterize all flat-foldable crease patterns. Therefore, so far, we have to enumerate and check flat-foldability one by one using computer. In this study, we develop the first algorithm for the above stated problem by combining these results in a nontrivial way and show its analysis of efficiency.

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.

For subgraph enumeration problems, very efficient algorithms have been proposed whose time complexities are far smaller than the number of subgraphs. Although the number of subgraphs can exponentially increase with the input graph size, these algorithms exploit compressed representations to output and maintain enumerated subgraphs compactly so as to reduce the time and space complexities. However, they are designed for enumerating only some specific types of subgraphs, e.g., paths or trees. In this paper, we propose an algorithm framework, called the frontier-based search, which generalizes these specific algorithms without losing their efficiency. Our frontier-based search will be used to resolve various practical problems that include constrained subgraph enumeration.

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.

In a rooted graph, a vertex is designated as its root. An outerplanar graph is represented by a plane embedding such that all vertices appear along its outer boundary. Two different plane embeddings of a rooted outerplanar graphs are called symmetric copies. Given integers n ≥ 3 and g ≥ 3, we give an O(n)-space and O(1)-time delay algorithm that generates all biconnected rooted outerplanar graphs with exactly n vertices such that the size of each inner face is at most g without delivering two symmetric copies of the same graph.