The most famous silhouette puzzle is the tangram, which originated in China more than two centuries ago. From around the same time, there is a similar Japanese puzzle called Sei Shonagon Chie no Ita. Both are derived by cutting a square of material with straight incisions into seven pieces of varying shapes, and each can be decomposed into sixteen non-overlapping identical right isosceles triangles. It is known that the pieces of the tangram can form thirteen distinct convex polygons. We first show that the Sei Shonagon Chie no Ita can form sixteen. Therefore, in a sense, the Sei Shonagon Chie no Ita is more expressive than the tangram. We also propose more expressive patterns built from the same 16 identical right isosceles triangles that can form nineteen convex polygons. There exist exactly four sets of seven pieces that can form nineteen convex polygons. We show no set of seven pieces can form at least 20 convex polygons, and demonstrate that eleven pieces made from sixteen identical isosceles right triangles are necessary and sufficient to form 20 convex polygons. Moreover, no set of six pieces can form nineteen convex polygons.

Last year, a new notion of rep-cube was proposed. A rep-cube is a polyomino that is a net of a cube, and it can be divided into some polyominoes such that each of them can be folded into a cube. This notion was inspired by the notions of polyomino and rep-tile, which were introduced by Solomon W. Golomb. It was proved that there are infinitely many distinct rep-cubes. In this paper, we investigate this new notion and show further results.

We study a new reconfiguration problem inspired by classic mechanical puzzles: a colored token is placed on each vertex of a given graph; we are also given a set of distinguished cycles on the graph. We are tasked with rearranging the tokens from a given initial configuration to a final one by using cyclic shift operations along the distinguished cycles. We call this a cyclic shift puzzle. We first investigate a large class of graphs, which generalizes several classic cyclic shift puzzles, and we give a characterization of which final configurations can be reached from a given initial configuration. Our proofs are constructive, and yield efficient methods for shifting tokens to reach the desired configurations. On the other hand, when the goal is to find a shortest sequence of shifting operations, we show that the problem is NP-hard, even for puzzles with tokens of only two different colors.

In this paper, we investigate computational complexity of pipe puzzles. A pipe puzzle is a kind of tiling puzzle; the input is a set of cards, and a part of a pipe is drawn on each card. For a given set of cards, we arrange them and connect the pipes. We have to connect all pipes without creating any local loop. While ordinary tiling puzzles, like jigsaw puzzles, ask to arrange the tiles with local consistency, pipe puzzles ask to join all pipes. We first show that the pipe puzzle is NP-complete in general even if the goal shape is quite restricted. We also investigate restricted cases and show some polynomial-time algorithms.

We investigate connected proper interval graphs without vertex labels. We first give the number of connected proper interval graphs of n vertices. Using this result, a simple algorithm that generates a connected proper interval graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected proper interval graphs is proposed. The algorithm is based on reverse search, and it outputs each connected proper interval graph in (O)1 time.

The maximal linear forest problem is to find, given a graph G = (V, E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2n) timeusing n4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5n) time using n3 processors on an EREW PRAM. The results put the problem in the class NC.

PREIMAGE CONSTRUCTION problem by Kratsch and Hemaspaandra naturally arose from the famous graph reconstruction conjecture. It deals with the algorithmic aspects of the conjecture. We present an O(n8) time algorithm for PREIMAGE CONSTRUCTION on permutation graphs and an O(n4(n+m)) time algorithm for PREIMAGE CONSTRUCTION on distance-hereditary graphs, where n is the number of graphs in the input, and m is the number of edges in a preimage. Since each graph of the input has n-1 vertices and O(n2) edges, the input size is O(n3) (, or O(nm)). There are polynomial time isomorphism algorithms for permutation graphs and distance-hereditary graphs. However the number of permutation (distance-hereditary) graphs obtained by adding a vertex to a permutation (distance-hereditary) graph is generally exponentially large. Thus exhaustive checking of these graphs does not achieve any polynomial time algorithm. Therefore reducing the number of preimage candidates is the key point.

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.

Folding an m×n square grid pattern along the edges of a grid is called map folding. We consider a decision problem in terms of whether a partial overlapping order of the squares aligning on the boundary of an m×n map is valid in a particular fold model called simple fold. This is a variation of the decision problem of valid total orders of the map in a simple fold model. We provide a linear-time algorithm to solve this problem, by defining an equivalence relation and computing the folding sequence sequentially, either uniquely or representatively.

Secure sets and defensive alliances in graphs are studied. They are sets of vertices that are safe in some senses. In this paper, we first present a fixed-parameter algorithm for finding a small secure set, whose running time is much faster than the previously known one. We then present improved bound on the smallest sizes of defensive alliances and secure sets for hypercubes. These results settle some open problems paused recently.

Longest path problem is a problem for finding a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, there are few known graph classes such that the longest path problem can be solved efficiently. Polynomial time algorithms for finding a longest cycle and a longest path in a Ptolemaic graph are proposed. Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. The algorithms use the dynamic programming technique on a laminar structure of cliques, which is a recent characterization of Ptolemaic graphs.

This paper introduces the maximization version of the k-path vertex cover problem, called the MAXIMUM K-PATH VERTEX COVER problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.

The Voronoi game is a two-person perfect information game modeling a competitive facility location. The original version of the game is played on a continuous domain. Only two special cases (1-dimensional case and 1-round case) have been extensively investigated. Recently, the discrete Voronoi game of which the game arena is given as a graph was introduced. In this note, we give a complete analysis of the discrete Voronoi game on a path. There are drawing strategies for both the first and the second players, except for some trivial cases.

In this paper, we introduce a path embedding problem inspired by the well-known hydrophobic-polar (HP) model of protein folding. A graph is said bicolored if each vertex is assigned a label in the set {red, blue}. For a given bicolored path P and a given bicolored graph G, our problem asks whether we can embed P into G in such a way as to match the colors of the vertices. In our model, G represents a protein's “blueprint,” and P is an amino acid sequence that has to be folded to form (part of) G. We first show that the bicolored path embedding problem is NP-complete even if G is a rectangular grid (a typical scenario in protein folding models) and P and G have the same number of vertices. By contrast, we prove that the problem becomes tractable if the height of the rectangular grid G is constant, even if the length of P is independent of G. Our proof is constructive: we give a polynomial-time algorithm that computes an embedding (or reports that no embedding exists), which implies that the problem is in XP when parameterized according to the height of G. Additionally, we show that the problem of embedding P into a rectangular grid G in such a way as to maximize the number of red-red contacts is NP-hard. (This problem is directly inspired by the HP model of protein folding; it was previously known to be NP-hard if G is not given, and P can be embedded in any way on a grid.) Finally, we show that, given a bicolored graph G, the problem of constructing a path P that embeds in G maximizing red-red contacts is Poly-APX-hard.

Logical matrices are binary matrices often used to represent relations. In the map folding problem, each folded state corresponds to a unique partial order on the set of squares and thus could be described with a logical matrix. The logical matrix representation is powerful than graphs or other common representations considering its association with category theory and homology theory and its generalizability to solve other computational problems. On the application level, such representations allow us to recognize map folding intuitively. For example, we can give a precise mathematical description of a folding process using logical matrices so as to solve problems like how to represent the up-and-down relations between all the layers according to their adjacency in a flat-folded state, how to check self-penetration, and how to deduce a folding process from a given order of squares that is supposed to represent a folded state of the map in a mathematical and natural manner. In this paper, we give solutions to these problems and analyze their computational complexity.

This paper deals with a popular puzzle known as Hi-Q. The puzzle is generalized: the board is extended to the size nn, an initial position of the puzzle is given, and a place is given on which only one token is finally placed. The complexity of the generalized Hi-Q is proved NP-complete.

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.

Shakashaka is a pencil-and-paper puzzle proposed by Guten and popularized by the Japanese publisher Nikoli (like Sudoku). We determine the computational complexity by proving that Shakashaka is NP-complete, and furthermore that counting the number of solutions is #P-complete. Next we formulate Shakashaka as an integer-programming (IP) problem, and show that an IP solver can solve every instance from Nikoli's website within a second.

A rep-cube is a polyomino that is a net of a cube, and it can be divided into some polyominoes such that each of them can be folded into a cube. This notion was invented in 2017, which is inspired by the notions of polyomino and rep-tile, which were introduced by Solomon W. Golomb. A rep-cube is called regular if it can be divided into the nets of the same area. A regular rep-cube is of order k if it is divided into k nets. Moreover, it is called uniform if it can be divided into the congruent nets. In this paper, we focus on these special rep-cubes and solve several open problems.