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Structural and Combinatorial Properties of 2-swap Word Permutation Graphs

Duncan Adamson, Nathan Flaherty, Igor Potapov, Paul G. Spirakis

TL;DR

This work studies the configuration graph $G(P)$ whose vertices are all words with a fixed Parikh vector $P$ and edges correspond to a single 2-swap operation. It proves an exact diameter $D = n - \max_{i \in \Sigma} P[i]$, characterizes the clique structure with $\\omega(G(P)) = \max_i P[i] + 1$, and establishes that $G(P)$ admits a Hamiltonian path starting at every vertex. The authors provide constructive enumeration algorithms that generate a Hamiltonian path via swaps, offering $O(n \log n)$ preprocessing and $O(\log n)$ delay for binary alphabets (and $O(\\sigma \log n)$ for general alphabets). These results contribute to the theory of combinatorial Gray codes on multiset permutations and have implications for efficient space exploration and enumeration tasks in related domains. The work also lays groundwork for extensions to $k$-swap variants and connections to combinatorial structures such as CSP and necklaces.

Abstract

In this paper, we study the graph induced by the $\textit{2-swap}$ permutation on words with a fixed Parikh vector. A $2$-swap is defined as a pair of positions $s = (i, j)$ where the word $w$ induced by the swap $s$ on $v$ is $v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$. With these permutations, we define the $\textit{Configuration Graph}$, $G(P)$ defined over a given Parikh vector. Each vertex in $G(P)$ corresponds to a unique word with the Parikh vector $P$, with an edge between any pair of words $v$ and $w$ if there exists a swap $s$ such that $v \circ s = w$. We provide several key combinatorial properties of this graph, including the exact diameter of this graph, the clique number of the graph, and the relationships between subgraphs within this graph. Additionally, we show that for every vertex in the graph, there exists a Hamiltonian path starting at this vertex. Finally, we provide an algorithm enumerating these paths from a given input word of length $n$ with a delay of at most $O(\log n)$ between outputting edges, requiring $O(n \log n)$ preprocessing.

Structural and Combinatorial Properties of 2-swap Word Permutation Graphs

TL;DR

This work studies the configuration graph whose vertices are all words with a fixed Parikh vector and edges correspond to a single 2-swap operation. It proves an exact diameter , characterizes the clique structure with , and establishes that admits a Hamiltonian path starting at every vertex. The authors provide constructive enumeration algorithms that generate a Hamiltonian path via swaps, offering preprocessing and delay for binary alphabets (and for general alphabets). These results contribute to the theory of combinatorial Gray codes on multiset permutations and have implications for efficient space exploration and enumeration tasks in related domains. The work also lays groundwork for extensions to -swap variants and connections to combinatorial structures such as CSP and necklaces.

Abstract

In this paper, we study the graph induced by the permutation on words with a fixed Parikh vector. A -swap is defined as a pair of positions where the word induced by the swap on is . With these permutations, we define the , defined over a given Parikh vector. Each vertex in corresponds to a unique word with the Parikh vector , with an edge between any pair of words and if there exists a swap such that . We provide several key combinatorial properties of this graph, including the exact diameter of this graph, the clique number of the graph, and the relationships between subgraphs within this graph. Additionally, we show that for every vertex in the graph, there exists a Hamiltonian path starting at this vertex. Finally, we provide an algorithm enumerating these paths from a given input word of length with a delay of at most between outputting edges, requiring preprocessing.
Paper Structure (9 sections, 18 theorems, 2 equations, 2 figures, 2 algorithms)

This paper contains 9 sections, 18 theorems, 2 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Basic Properties of the Configuration Graph
  5. Hamiltonicity
  6. Binary Alphabets
  7. General Alphabets.
  8. Conclusion
  9. Acknowledgements

Key Result

Lemma 1

Given a Parikh vector $P$ with associated configuration graph $G(P)$, each vertex $v \in V(P)$ belongs to $\sum_{j \in \Sigma}\prod_{i \in \Sigma \setminus \{j\}} P[i]$ maximal cliques, with the size of each such clique being in $\{P[i]+1 \mid i \in \Sigma\}$.

Figures (2)

  • Figure 1: The configuration graph $G(3,2)$ with Hamiltonian path shown in red.
  • Figure 2: The graph $G(aaabbc,bcbaaa)$ with the edge $(i,j)$ labelled by the symbol $w[i] (= v[j])$.

Theorems & Definitions (22)

  • definition 1: 2-swap
  • Example 1
  • definition 2
  • Lemma 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 2
  • Theorem 3
  • Lemma 4: Upper Bound of Diameter
  • ...and 12 more