# Complexity of Hamiltonian Cycle Reconfiguration

## Abstract

**:**

## 1. Introduction

#### 1.1. Our Contribution

#### 1.2. Notation

## 2. PSPACE-Completeness

#### 2.1. Nondeterministic Constraint Logic

- for every vertex $v\in A$, the sum of weights of in-coming edges of v is at least 2, and
- every vertex of B has one or two in-coming edges, but at most one vertex of B has two in-coming edges.

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

#### 2.2. Reduction

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Theorem**

**2.**

#### 2.3. Strongly Chordal Split Graphs

**Theorem**

**3.**

## 3. Canonical Hamiltonian Cycles

#### 3.1. Unit Interval Graphs

**Theorem**

**4**

**Theorem**

**5**

**Theorem**

**6.**

**Lemma**

**3.**

**Proof.**

**Proof**

**of**

**Theorem**

**6.**

- ${C}_{i}$ contains the edges on ${C}_{t}$ induced by $\{{v}_{0},{v}_{1},\dots ,{v}_{i}\}$,
- ${C}_{i}$ is obtained from ${C}_{i-1}$ by at most one switch.

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

#### 3.2. Bipartite Permutation Graphs

**Theorem**

**7**

**Theorem**

**8**

**Theorem**

**9.**

**Proof.**

- ${C}_{i}$ contains the edges on ${C}_{t}$ induced by $\{{v}_{0},{v}_{1},\dots ,{v}_{i}\}$, where ${v}_{0}={u}_{0}$, ${v}_{1}={w}_{0}$, ${v}_{2}={u}_{1}$, ${v}_{3}={w}_{1}$, …, ${v}_{n-2}={u}_{p-1}$, ${v}_{n-1}={w}_{p-1}$;
- ${C}_{i}$ is obtained from ${C}_{i-1}$ by at most one switch.

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**All the possible orientations of edges incident to an AND vertex, where (blue) thick arrows denote the edges with weight 2, and (red) thin arrows denote the edges with weight 1. Each dotted circle represents a possible orientation of the edges, and two circles are joined by an arrow if one is obtained from the other by reversing the direction of a single edge.

**Figure 2.**The reduction from the nondeterministic constraint logic problem to the problem $\Pi $. White points denote the vertices of A, and gray points denote the vertices of B. Thick (blue) lines denote the edges with weight 2, and thin (red) lines denote the edges with weight 1.

**Figure 4.**All the possible configurations of a Hamiltonian cycle passing through gadgets. The edges on the cycle are indicated by thick lines, but the ears are omitted; the edges out of the cycle are indicated by dotted lines. Each dotted square represents a possible configuration, and two squares are joined by an arrow if one is obtained from the other by a single switch.

**Figure 5.**(

**a**) a legal orientation of the problem $\Pi $. White points denote the vertices of A, and gray points denote the vertices of B. Thick (blue) lines denote the edges with weight 2, and thin (red) lines denote the edges with weight 1; (

**b**) the Hamiltonian cycle obtained from the legal orientation in Figure 5a. We take the configuration ${S}_{3}$ for the gadget for ${a}_{2}$, since the edges of ${a}_{2}$ are oriented as ${f}_{3}$ in Figure 5a. Notice that, when we replace the configuration from ${S}_{3}$ to ${S}_{4}$, we have two cycles.

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**MDPI and ACS Style**

Takaoka, A.
Complexity of Hamiltonian Cycle Reconfiguration. *Algorithms* **2018**, *11*, 140.
https://doi.org/10.3390/a11090140

**AMA Style**

Takaoka A.
Complexity of Hamiltonian Cycle Reconfiguration. *Algorithms*. 2018; 11(9):140.
https://doi.org/10.3390/a11090140

**Chicago/Turabian Style**

Takaoka, Asahi.
2018. "Complexity of Hamiltonian Cycle Reconfiguration" *Algorithms* 11, no. 9: 140.
https://doi.org/10.3390/a11090140