TY - GEN

T1 - Structural and combinatorial properties of 2-swap word permutation graphs

AU - Adamson, Duncan

AU - Flaherty, Nathan

AU - Potapov, Igor

AU - Spirakis, Paul G.

N1 - Funding: This work is supported by the Leverhulme Research Centre for Functional Materials Design and EPSRC grants EP/P02002X/1, EP/R018472/1.

PY - 2024/3/6

Y1 - 2024/3/6

N2 - In this paper, we study the graph induced by the 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]⋯v[i-1]v[j]v[i+1]⋯v[j-1]v[i]v[j+1]⋯v[n]. With these permutations, we define the Configuration Graph, G(P) for 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∘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.

AB - In this paper, we study the graph induced by the 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]⋯v[i-1]v[j]v[i+1]⋯v[j-1]v[i]v[j+1]⋯v[n]. With these permutations, we define the Configuration Graph, G(P) for 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∘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.

UR - https://doi.org/10.1007/978-3-031-55601-2

U2 - 10.1007/978-3-031-55601-2_5

DO - 10.1007/978-3-031-55601-2_5

M3 - Conference contribution

AN - SCOPUS:85188729365

SN - 9783031556005

T3 - Lecture notes in computer science

SP - 61

EP - 76

BT - LATIN 2024 - Theoretical informatics

A2 - Soto, José A.

A2 - Wiese, Andreas

PB - Springer

CY - Cham

T2 - 16th Latin American Symposium on Theoretical Informatics, LATIN 2042

Y2 - 18 March 2024 through 22 March 2024

ER -