arXiv subjects

Tadashi Sakuma

Publications and source records attributed to Tadashi Sakuma.

12 recordsLinked to original sources

arXiv2026

Structural Origins of Cubic Complexity in Pebble Motion

The pebble motion problem (PMP) asks whether one configuration of labeled pebbles on a graph can be transformed into another by moving pebbles to adjacent unoccupied vertices. It is a fundamental model of graph reconfiguration and is closely related to multi-agent path finding (MAPF). A central open problem since Kornhauser, Miller, and Spirakis (FOCS 1984) is to understand the origin of the classical $Θ(N^3)$ worst-case behavior. While it is known that every feasible instance on an $N$-vertex graph admits a solution sequence of length $\Ord(N^3)$, it has remained unclear which instances actually require cubic complexity. In this paper, we resolve the long-standing complexity of the pebble motion problem on trees. We show that every feasible instance on an $N$-vertex tree admits a solution sequence of length $\Ord(N^2 \log N)$, computable by an output-sensitive algorithm. Since a lower bound of $Ω(N^2)$ is known, this establishes that the $Θ(N^3)$ phenomenon does not occur on trees and nearly closes the gap $Ω(N^2)\le \OPT(N)\le \Ord(N^3)$ up to a logarithmic factor. Building on this result, we extend our approach to general graphs by applying the tree algorithm to breadth-first spanning trees. This yields an efficient framework that produces $o(N^3)$-length solution sequences for a broad class of instances, including the classical square-grid example, where we recover the $\Ord(N^{3/2})$ bound observed by Kornhauser, Miller, and Spirakis. Finally, by analyzing the behavior of this algorithm, we obtain strong structural restrictions governing when $Θ(N^3)$ complexity can arise. We show that such behavior is possible only under highly constrained conditions, specifically when $Θ(N)$ degree-two vertices lie on cycles of length $Θ(N)$, with each cycle being the shortest containing the corresponding vertex.

math.CO↗

arXiv2025

Universal graph series, chromatic functions, and their index theory

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.

math.CO↗

arXiv2024

The Tutte polynomials of genus $g$

In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 95(10) 111-113], the authors introduce the concept of the Tutte polynomials of genus $g$ and announce that each matroid $M$ can be reconstructed from its Tutte polynomial of genus $|\mathcal{B}(M)|$, where $\mathcal{B}(M)$ denotes the family of bases of $M$. In that paper, we also announced that, for all $g$, there exist inequivalent matroids that have the same Tutte polynomial of genus $g$. In this paper, we prove these theorems.

math.CO↗

arXiv2021

On the average hitting times of the squares of cycles

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.

math.CO↗

arXiv2021

Pebble Exchange Group of Graphs

A graph puzzle ${\rm Puz}(G)$ of a graph $G$ is defined as follows. A configuration of ${\rm Puz}(G)$ is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some input graph $G$. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations $f$ and $g$, we say that $f$ is equivalent to $g$ if $f$ can be transformed into $g$ by a finite sequence of moves. Let ${\rm Aut}(G)$ be the automorphism group of $G$, and let ${\rm 1}_G$ be the unit element of ${\rm Aut}(G)$. The pebble exchange group of $G$, denoted by ${\rm Peb}(G)$, is defined as the set of all automorphisms $f$ of $G$ such that ${\rm 1}_G$ and $f$ are equivalent to each other. In this paper, some basic properties of ${\rm Peb}(G)$ are studied. Among other results, it is shown that for any connected graph $G$, all automorphisms of $G$ are contained in ${\rm Peb}(G^2)$, where $G^2$ is a square graph of $G$.

cs.DM↗

arXiv2020

Tutte polynomial, complete invariant, and theta series

In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let L_1 and L_2 be two non-isomorphic lattices. We state that L_1 and L_2 are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof.~Conway's book The Sensual (Quadratic) Form with the title "Can You Hear the Shape of a Lattice?" In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials.

math.CO↗

arXiv2020

Stable structure on safe set problems in vertex-weighted graphs

Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $\mathrm{s}(G,w)$ and connected weighted safe number $\mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $\mathrm{s}(G,w)=\mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.

math.CO↗

arXiv2019

A generalization of the Tutte polynomials

In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.

math.CO↗

arXiv2018

On the weighted safe set problem on paths and cycles

Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $s(G,w)$ and connected weighted safe number $cs(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. It is easy to see that for any pair $(G,w)$, ${s}(G,w) \le {cs}(G,w)$ by their definitions. In this paper, we discuss the possible equality when $G$ is a path or a cycle. We also give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32 (2011)] concerning subgraph component polynomials for cycles and complete graphs.

math.CO↗

arXiv2014

On the balanced decomposition number

A {\em balanced coloring} of a graph $G$ means a triple $\{P_1,P_2,X\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \uplus P_2 \uplus X$ and $|P_1|=|P_2|$. A {\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \uplus P_2 \uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \uplus \cdots \uplus V_r$ (for some $r$) such that, for every $i \in \{1,\cdots,r\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \cap P_1| = |V_i \cap P_2|$. Then the {\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ such that, for every balanced coloring $V(G)=P_1 \uplus P_2 \uplus X$ of $G$, there exists a balanced decomposition $V(G)=V_1 \uplus \cdots \uplus V_r$ whose every element $V_i (i=1, \cdots, r)$ has at most $s$ vertices. S. Fujita and H. Liu [\/SIAM J. Discrete Math. 24, (2010), pp. 1597--1616\/] proved a nice theorem which states that the balanced decomposition number of a graph $G$ is at most $3$ if and only if $G$ is $\lfloor\frac{|V(G)|}{2}\rfloor$-connected. Unfortunately, their proof is lengthy (about 10 pages) and complicated. Here we give an immediate proof of the theorem. This proof makes clear a relationship between balanced decomposition number and graph matching.

math.CO↗

arXiv2012

Agent Arrangement Problem

An {\em arrangement} of an ordered pair $(G_A, G_M)$ of graphs is defined as a function $f$ from $V(G_A)$ to $V(G_M)$ such that, for each vertex $c$ of $G_M$, the vertex-set $f^{-1}(c)$ of $G_A$ either is $\emptyset$ (the case when $c \not\in f(V(G_A))$) or induces a connected subgraph of $G_A$ and that the family $\{f^{-1}(y) : y \in V(G_M), f^{-1}(y) \neq \emptyset\}$ is a partition of $V(G_A)$. Let $f$ be an arrangement of $(G_A, G_M)$, let $pq$ be an edge of $G_M$ and let $U$ be a subset of $f^{-1}(p)$ such that each of the three graphs $G_A[U]$, $G_A[f^{-1}(p)\setminus U]$ and $G_A[f^{-1}(q)\cup U]$ is ether connected or $\emptyset$ and that $\big(f^{-1}(p)\cup f^{-1}(q) \big) \setminus U \neq \emptyset$. A {\em transfer} of $U$ from $p$ to $q$ is defined as the modification $f^{\prime}$ of $f$ such that $f^{\prime}(x):=f(x)$ for every $ x \notin U$ and $f^{\prime}(u):=q$ for every $u \in U$. Two arrangements $f$ and $g$ of $(G_A, G_M)$ are called {\em t-equivalent} if they can be transformed into each other by a finite sequence of transfers. An ordered pair $(G_A, G_M)$ of graphs is called {\em almighty} if every two arrangements of the pair $(G_A, G_M)$ are t-equivalent. In this study, we consider the following two decision problems. [{\bf (P1)}]{For a given pair of arrangements $f$ and $g$ of a given ordered pair $(G_A,G_M)$ of graphs, decide whether $f$ is t-equivalent to $g$ or not.} [{\bf (P2)}]{For a given ordered pair $(G_A,G_M)$ of graphs, decide whether the pair $(G_A,G_M)$ is almighty or not.} We show an $\Od(|E(G_A)|+(|V(G_M)|+|E(G_A)|)|V(G_A)|)$-time algorithm for {\bf (P1)}, and prove the $\co\np$-completeness of {\bf (P2)}.

math.CO↗

arXiv2012

On ideal minimally non-packing clutters

We consider the following conjecture proposed by Cornuéjols, Guenin and Margot: every ideal minimally non-packing clutter has a transversal of size 2. For a clutter C, the tilde clutter is the set of hyperedges of C which intersect any minimum transversal in exactly one element. We divide the (non-)existence problem of an ideal minimally non-packing clutter D into two steps. In the first step, we give necessary conditions for C = the tilde clutter of D when a clutter D is an ideal minimally non-packing clutter. In the second step, for a clutter C satisfying the conditions in the first step, we consider whether C has an ideal minimally non-packing clutter D with C= the tilde clutter of D. We show that the clutter of a combinatorial affine plane satisfies the conditions in the first step. Moreover, we show that the clutter of a combinatorial affine plane does not have any ideal minimally non-packing clutter of blocking number at least 3.

math.CO↗