Structural and combinatorial properties of 2-swap word permutation graphs
Abstract
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.
Citation
Adamson , D , Flaherty , N , Potapov , I & Spirakis , P G 2024 , Structural and combinatorial properties of 2-swap word permutation graphs . in J A Soto & A Wiese (eds) , LATIN 2024 - Theoretical informatics : 16th Latin American Symposium, Puerto Varas, Chile, March 18-22, 2024, Proceedings, Part II . Lecture notes in computer science , vol. 14579 , Springer , Cham , pp. 61-76 , 16th Latin American Symposium on Theoretical Informatics, LATIN 2042 , Puerto Varas , Chile , 18/03/24 . https://doi.org/10.1007/978-3-031-55601-2_5 conference
Publication
LATIN 2024 - Theoretical informatics
ISSN
0302-9743Type
Conference item
Description
Funding: This work is supported by the Leverhulme Research Centre for Functional Materials Design and EPSRC grants EP/P02002X/1, EP/R018472/1.Collections
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