arXiv subjects

Takashi Horiyama

Publications and source records attributed to Takashi Horiyama.

23 records · Page 2Linked to original sources

arXiv2019

Implicit Enumeration of Topological-Minor-Embeddings and Its Application to Planar Subgraph Enumeration

Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.

cs.DS↗

arXiv2019

Decomposing a Graph into Unigraphs

Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these subclasses are well studied in the literature. Nevertheless, there are few results on superclasses of unigraphs. In this paper, we introduce two types of generalizations of unigraphs: $k$-unigraphs and $k$-strong unigraphs. We say that a graph $G$ is a $k$-unigraph if $G$ can be partitioned into $k$ unigraphs. $G$ is a $k$-strong unigraph if not only each subgraph is a unigraph but also the whole graph can be uniquely determined up to isomorphism, by using the degree sequences of all the subgraphs in the partition. We describe a relation between $k$-strong unigraphs and the subgraph isomorphism problem. We show some properties of $k$-(strong) unigraphs and algorithmic results on calculating the minimum $k$ such that a graph $G$ is a $k$-(strong) unigraph. This paper will open many other research topics.

cs.DS↗

arXiv2018

Rigid Foldability is NP-Hard

In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldability from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matter and reconfigurable robots.

cs.CG↗

arXiv2018

Enumerating Cryptarithms Using Deterministic Finite Automata

A cryptarithm is a mathematical puzzle where given an arithmetic equation written with letters rather than numerals, a player must discover an assignment of numerals on letters that makes the equation hold true. In this paper, we propose a method to construct a DFA that accepts cryptarithms that admit (unique) solutions for each base. We implemented the method and constructed a DFA for bases $k \le 7$. Those DFAs can be used as complete catalogues of cryptarithms,whose applications include enumeration of and counting the exact numbers $G_k(n)$ of cryptarithm instances with $n$ digits that admit base-$k$ solutions. Moreover, explicit formulas for $G_2(n)$ and $G_3(n)$ are given.

cs.FL↗

arXiv2018

Swapping Colored Tokens on Graphs

We investigate the computational complexity of the following problem. We are given a graph in which each vertex has an initial and a target color. Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number of swaps so that the current and target colors agree at each vertex. When the colors are chosen from {1,2,...,c}, we call this problem c-Colored Token Swapping since the current color of a vertex can be seen as a colored token placed on the vertex. We show that c-Colored Token Swapping is NP-complete for c = 3 even if input graphs are restricted to connected planar bipartite graphs of maximum degree 3. We then show that 2-Colored Token Swapping can be solved in polynomial time for general graphs and in linear time for trees. Besides, we show that, the problem for complete graphs is fixed-parameter tractable when parameterized by the number of colors, while it is known to be NP-complete when the number of colors is unbounded.

cs.DS↗