arXiv subjects

Duc A. Hoang

Publications and source records attributed to Duc A. Hoang.

14 recordsLinked to original sources

arXiv2026

Token-sliding realizability for complements, Cartesian-products, and grid graph families

For an integer $k\ge 0$ and a graph $G$, the \emph{token-sliding reconfiguration graph $\mathsf{TS}_k(G)$} has the independent $k$-sets of $G$ as vertices. Two vertices are adjacent if one token can slide along an edge of $G$ and the resulting $k$-set is still independent. We study the following realizability problem: for fixed $k\ge 2$, which graphs are isomorphic to $\mathsf{TS}_k(G)$ for some graph $G$? This inverse viewpoint asks which abstract state spaces can occur exactly under a local token rule. We give positive realizability results for the complement targets $\overline{K_n}$, $\overline{K_{m,n}}$, and $\overline{K_n-e}$, and we determine sharp cutoffs for complements of paths and cycles. We also prove a product formula for token-sliding graphs of disjoint unions and apply it to Cartesian products of complete graphs, paths, and cycles. For every grid $Γ_{m,n}=P_m\square P_n$ with $2\le m\le n$, we realize $Γ_{m,n}$ at token value $m+n-2$ and at every token value $k\ge 4$. At small token values, we prove that $C_4\square C_n$ is not a $\mathsf{TS}_2$-graph for $n\ge 4$, classify ladders $Γ_{2,n}$, and settle the first non-ladder grid: for $k\ge 2$, $Γ_{3,3}$ is realizable if and only if $k\ge 4$.

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arXiv2026

On Realizing Reconfiguration Graphs of Cliques

For a graph $H$ and an integer $k\ge 1$, the \emph{Token Sliding reconfiguration graph} $\mathsf{TS}_k(H)$ and the \emph{Token Jumping reconfiguration graph} $\mathsf{TJ}_k(H)$ have as vertices the $k$-cliques of $H$, with two vertices adjacent when one clique is obtained from the other by replacing one vertex with an adjacent non-member, and respectively by an arbitrary non-member. For a target graph $G$, we study the feasibility sets $\mathcal{K}^{\mathsf{TS}}(G)$ and $\mathcal{K}^{\mathsf{TJ}}(G)$, consisting of all integers $k$ for which $G$ is isomorphic to $\mathsf{TS}_k(H)$ and $\mathsf{TJ}_k(H)$, respectively, for some graph $H$. We determine the exact feasibility sets for complete graphs, paths, cycles, complete bipartite graphs, book graphs, friendship graphs, and their complements, and give complete classifications for all Johnson graphs.

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arXiv2026

Distance Recoloring

Reconfiguration problems ask whether one feasible solution can be transformed into another by a sequence of local moves while maintaining feasibility throughout. For integers $d \geq 1$ and $k \geq d+1$, the Distance Coloring problem asks if a given graph $G$ has a $(d, k)$-coloring, i.e., a coloring of the vertices of $G$ by $k$ colors such that any two vertices within distance $d$ from each other have different colors. For ordinary proper colorings ($d=1$), the $k$-Coloring Reconfiguration problem is polynomial-time solvable for $k\le 3$ [Cereceda, van den Heuvel, and Johnson, J. Graph Theory 67(1):69--82, 2011] but is $\mathsf{PSPACE}$-complete for every fixed $k\ge 4$, even on bipartite graphs [Bonsma and Cereceda, Theor. Comput. Sci. 410(50):5215--5226, 2009]. In this work, we initiate a study of the distance-$d$ analogue, for $d \geq 2$. We show that even for planar, bipartite, and $2$-degenerate graphs, $(d, k)$-Coloring Reconfiguration remains $\mathsf{PSPACE}$-complete for every $d \geq 3$ via a reduction from the well-known Sliding Tokens problem. Our construction uses $k = k_0 + 2 + n(\lceil d/2\rceil-1)$ colors on instances of size $n$, where $k_0\in\{3d+3,3d+6\}$ (depending on the parity of $d$). For $d = 2$, the same reduction scheme can be adapted to show that the problem is $\mathsf{PSPACE}$-complete on planar and $2$-degenerate graphs with same values of $k$. Additionally, on split graphs, there is an interesting dichotomy: the problem is $\mathsf{PSPACE}$-complete when $d = 2$ and $k$ is large but can be solved efficiently when $d \geq 3$ and $k \geq d+1$. For chordal graphs, we show that the problem is $\mathsf{PSPACE}$-complete for even values of $d \geq 2$. Finally, we design a quadratic-time algorithm to solve the problem on paths for any $d \geq 2$ and $k \geq d+1$.

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arXiv2026

Directed Token Sliding

Reconfiguration problems involve determining whether two given configurations can be transformed into each other under specific rules. The Token Sliding problem asks whether, given two different set of tokens on vertices of a graph $G$, we can transform one into the other by sliding tokens step-by-step along edges of $G$ such that each resulting set of tokens forms an independent set in $G$. Recently, Ito et al. [MFCS 2022] introduced a directed variant of this problem. They showed that for general oriented graphs (i.e., graphs where no pair of vertices can have directed edges in both directions), the problem remains $\mathsf{PSPACE}$-complete, and is solvable in polynomial time on oriented trees. In this paper, we further investigate the Token Sliding problem on various oriented graph classes. We show that the problem remains $\mathsf{PSPACE}$-complete for oriented split graphs, bipartite graphs and bounded treewidth graphs. Additionally, we present polynomial-time algorithms for solving the problem on oriented cycles and cographs.

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arXiv2025

A Note on Reconfiguration Graphs of Cliques

In a reconfiguration setting, each clique of a graph $G$ is viewed as a set of tokens placed on vertices of $G$ such that no vertex has more than one token and any two tokens are adjacent. Three well-known reconfiguration rules have been studied in the literature: Token Jumping ($\mathsf{TJ}$), Token Sliding ($\mathsf{TS}$), and Token Addition/Removal ($\mathsf{TAR}$). Given a graph $G$ and a reconfiguration rule $\mathsf{R} \in \{\mathsf{TS}, \mathsf{TJ}, \mathsf{TAR}\}$, a reconfiguration graph of $k$-cliques of $G$, denoted by $\mathsf{R}_k(G)$, is the graph whose vertices are cliques of $G$ of size $k$ and two vertices are adjacent if one can be obtained from the other by applying $\mathsf{R}$ exactly once. In this paper, we initiate the study of structural properties of reconfiguration graphs of cliques, proving several interesting results primarily under $\mathsf{TS}$ and $\mathsf{TJ}$ rules. In particular, we establish a formula relating the clique number of $G$ and that of $\mathsf{TS}_k(G)$, and bound the chromatic number of $\mathsf{TS}_k(G)$ via that of an appropriate Johnson graph. Additionally, we present an algorithm to construct $\mathsf{TS}_{ω(G)-1}(G)$ from $\mathsf{TJ}_{ω(G)}(G)$ and derive structural properties of $\mathsf{TJ}_{ω(G)}(G)$ graphs, where $ω(G)$ denotes the clique number of $G$. Finally, we show that $\mathsf{TS}_k(G)$ is planar whenever $G$ is planar and establish bounds on the number of $3$- and $4$-cliques based on results concerning $\mathsf{TS}_k(G)$ graphs. In particular, we prove that any planar graph $G$ with $n$ vertices can contain at most $3n - 8$ triangles, which aligns with the classical bound on maximal planar graphs.

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arXiv2025

The Complexity of Distance-$r$ Dominating Set Reconfiguration

For a fixed integer $r \geq 1$, a distance-$r$ dominating set (D$r$DS) of a graph $G = (V, E)$ is a vertex subset $D \subseteq V$ such that every vertex in $V$ is within distance $r$ from some member of $D$. Given two D$r$DSs $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of D$r$DSs that transforms $D_s$ into $D_t$ (or vice versa) such that each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. The problem for $r = 1$ has been well-studied in the literature. We consider D$r$DSR for $r \geq 2$ under two well-known reconfiguration rules: Token Jumping ($\mathsf{TJ}$, which involves replacing a member of the current D$r$DS by a non-member) and Token Sliding ($\mathsf{TS}$, which involves replacing a member of the current D$r$DS by an adjacent non-member). It is known that under any of $\mathsf{TS}$ and $\mathsf{TJ}$, the problem on split graphs is $\mathtt{PSPACE}$-complete for $r = 1$. We show that for $r \geq 2$, the problem is in $\mathtt{P}$, resulting in an interesting complexity dichotomy. Along the way, we prove some non-trivial bounds on the length of a shortest reconfiguration sequence on split graphs when $r = 2$ which may be of independent interest. Additionally, we design a linear-time algorithm under $\mathsf{TJ}$ on trees. On the negative side, we show that D$r$DSR for $r \geq 1$ on planar graphs of maximum degree three and bounded bandwidth is $\mathtt{PSPACE}$-complete, improving the degree bound of previously known results. We also show that the known $\mathtt{PSPACE}$-completeness results under $\mathsf{TS}$ and $\mathsf{TJ}$ for $r = 1$ on bipartite graphs and chordal graphs can be extended for $r \geq 2$.

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arXiv2024

On the Complexity of Distance-$d$ Independent Set Reconfiguration

For a fixed positive integer $d \geq 2$, a distance-$d$ independent set (D$d$IS) of a graph is a vertex subset whose distance between any two members is at least $d$. Imagine that there is a token placed on each member of a D$d$IS. Two D$d$ISs are adjacent under Token Sliding ($\mathsf{TS}$) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping ($\mathsf{TJ}$), the target vertex needs not to be adjacent to the original one. The Distance-$d$ Independent Set Reconfiguration (D$d$ISR) problem under $\mathsf{TS}/\mathsf{TJ}$ asks if there is a corresponding sequence of adjacent D$d$ISs that transforms one given D$d$IS into another. The problem for $d = 2$, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of D$d$ISR on different graphs under $\mathsf{TS}$ and $\mathsf{TJ}$ for any fixed $d \geq 3$. On chordal graphs, we show that D$d$ISR under $\mathsf{TJ}$ is in $\mathtt{P}$ when $d$ is even and $\mathtt{PSPACE}$-complete when $d$ is odd. On split graphs, there is an interesting complexity dichotomy: D$d$ISR is $\mathtt{PSPACE}$-complete for $d = 2$ but in $\mathtt{P}$ for $d=3$ under $\mathsf{TS}$, while under $\mathsf{TJ}$ it is in $\mathtt{P}$ for $d = 2$ but $\mathtt{PSPACE}$-complete for $d = 3$. Additionally, certain well-known hardness results for $d = 2$ on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for $d \geq 3$.

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arXiv2023

On Reconfiguration Graphs of Independent Sets under Token Sliding

An independent set of a graph $G$ is a vertex subset $I$ such that there is no edge joining any two vertices in $I$. Imagine that a token is placed on each vertex of an independent set of $G$. The $\mathsf{TS}$- ($\mathsf{TS}_k$-) reconfiguration graph of $G$ takes all non-empty independent sets (of size $k$) as its nodes, where $k$ is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph $G$: (1) Whether the $\mathsf{TS}_k$-reconfiguration graph of $G$ belongs to some graph class $\mathcal{G}$ (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If $G$ satisfies some property $\mathcal{P}$ (including $s$-partitedness, planarity, Eulerianity, girth, and the clique's size), whether the corresponding $\mathsf{TS}$- ($\mathsf{TS}_k$-) reconfiguration graph of $G$ also satisfies $\mathcal{P}$, and vice versa. Additionally, we give a decomposition result for splitting a $\mathsf{TS}_k$-reconfiguration graph into smaller pieces.

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arXiv2023

A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

We continue the study of token sliding reconfiguration graphs of independent sets initiated by the authors in an earlier paper (arXiv:2203.16861). Two of the topics in that paper were to study which graphs $G$ are token sliding graphs and which properties of a graph are inherited by a token sliding graph. In this paper we continue this study specializing on the case of when $G$ and/or its token sliding graph $\mathsf{TS}_k(G)$ is a tree or forest, where $k$ is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on $G$ for $\mathsf{TS}_k(G)$ to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a token sliding graph. For the first problem we give a forbidden subgraph characterization for the cases of $k=2,3$. For the second problem we show that for every $k$-ary tree $T$ there is a graph $G$ for which $\mathsf{TS}_{k+1}(G)$ is isomorphic to $T$. A number of other results are given along with a join operation that aids in the construction of $\mathsf{TS}_k(G)$-graphs.

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arXiv2022

TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the $k$-Path Vertex Cover Reconfiguration ($k$-PVCR) under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be $\mathtt{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, as a first step toward answering this question, for $k \geq 4$, we present a polynomial-time algorithm that solves $k$-PVCR under $\mathsf{TS}$ for caterpillars (i.e., trees formed by attaching leaves to a path).

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arXiv2022

Reconfiguring k-path vertex covers

A vertex subset $I$ of a graph $G$ is called a $k$-path vertex cover if every path on $k$ vertices in $G$ contains at least one vertex from $I$. The \textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} problem asks if one can transform one $k$-path vertex cover into another via a sequence of $k$-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of \textsc{$k$-PVCR} from the viewpoint of graph classes under the well-known reconfiguration rules: $\mathsf{TS}$, $\mathsf{TJ}$, and $\mathsf{TAR}$. The problem for $k=2$, known as the \textsc{Vertex Cover Reconfiguration (VCR)} problem, has been well-studied in the literature. We show that certain known hardness results for \textsc{VCR} on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for \textsc{$k$-PVCR}. In particular, we prove a complexity dichotomy for \textsc{$k$-PVCR} on general graphs: on those whose maximum degree is $3$ (and even planar), the problem is $\mathtt{PSPACE}$-complete, while on those whose maximum degree is $2$ (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for \textsc{$k$-PVCR} on trees under each of $\mathsf{TJ}$ and $\mathsf{TAR}$. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given $k$-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

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arXiv2019

Shortest Reconfiguration Sequence for Sliding Tokens on Spiders

Suppose that two independent sets $I$ and $J$ of a graph with $\vert I \vert = \vert J \vert$ are given, and a token is placed on each vertex in $I$. The Sliding Token problem is to determine whether there exists a sequence of independent sets which transforms $I$ into $J$ so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. It is one of the representative reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. For a yes-instance of a reconfiguration problem, finding a shortest reconfiguration sequence has a different aspect. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be $\mathsf{NP}$-hard to find a shortest sequence between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for spiders (i.e., trees having exactly one vertex of degree at least three).

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arXiv2018

Polynomial-Time Algorithms for Sliding Tokens on Cactus Graphs and Block Graphs

Given two independent sets $I, J$ of a graph $G$, and imagine that a token (coin) is placed at each vertex of $I$. The Sliding Token problem asks if one could transform $I$ to $J$ via a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors so that the resulting set of vertices where tokens are placed remains independent. This problem is $\mathsf{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded-treewidth. In this paper, we show that Sliding Token can be solved efficiently for cactus graphs and block graphs, and give upper bounds on the length of a transformation sequence between any two independent sets of these graph classes. Our algorithms are designed based on two main observations. First, all structures that forbid the existence of a sequence of token slidings between $I$ and $J$, if exist, can be found in polynomial time. A sufficient condition for determining no-instances can be easily derived using this characterization. Second, without such forbidden structures, a sequence of token slidings between $I$ and $J$ does exist. In this case, one can indeed transform $I$ to $J$ (and vice versa) using a polynomial number of token-slides.

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arXiv2014

Linear-Time Algorithm for Sliding Tokens on Trees

Suppose that we are given two independent sets $I_b$ and $I_r$ of a graph such that $|I_b|=|I_r|$, and imagine that a token is placed on each vertex in $I_b$. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms $I_b$ into $I_r$ so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between $I_b$ and $I_r$ whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

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