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In this work, the authors explore various properties and characteristics of word-representable graphs, parity graphs, and comparability graphs, while introducing the concept of permutation-representation number (prn) and prn-irreducible graphs.

## Word-Representable Graphs

The authors first demonstrate that the class of word-representable graphs is closed under split recomposition. This means that when two word-representable graphs are recomposed, the resulting graph also belongs to the same class. This finding expands our understanding of the behavior and composition of word-representable graphs.

Additionally, the authors determine the representation number of the graph obtained through recomposing two word-representable graphs. The representation number provides insights into the minimum number of words needed to represent a given graph. By identifying this number, researchers can better understand the complexity and structure of word-representable graphs.

## Parity Graphs

The authors establish that the class of parity graphs is word-representable. Parity graphs are a specialized type of graph where each vertex is assigned either an “even” or “odd” label. This result suggests a strong relationship between word representations and parity graphs, opening up potential avenues for further investigations into this correspondence.

## Composability of Comparability Graphs

The authors introduce a characteristic property that allows for the determination of whether the recomposition of comparability graphs results in another comparability graph. Comparability graphs are those that can be represented by a partial order, meaning that their vertices can be arranged in a linear order without creating any cycles or inconsistencies.

By identifying the conditions under which the recomposition of comparability graphs yields another comparability graph, we gain a deeper understanding of the underlying structure and constraints of comparability graphs. This finding could potentially have applications in optimization problems, network analysis, and various other fields that involve analyzing and manipulating comparability graphs.

## Permutation-Representation Number (prn)

In this work, the authors introduce the concept of permutation-representation number (prn) for comparability graphs. The prn of a graph represents the minimum number of permutations required to represent it. This measure provides insights into the complexity and the number of transformations needed to represent a given comparability graph.

By determining the prn of the resulting comparability graph obtained through the split recomposition of two prn-irreducible graphs, the authors contribute to our understanding of the compositional properties and complexity of these graphs. This knowledge can be useful in various applications such as algorithm design, network analysis, and data visualization.

## prn-Irreducible Graphs

The authors introduce a subclass of comparability graphs called prn-irreducible graphs. These graphs possess certain properties that make their permutations representation non-trivial. By providing a criterion for determining whether the split recomposition of two prn-irreducible graphs results in a comparability graph, the authors shed light on the behavior and characteristics of these specialized graphs.

Furthermore, by determining the prn of the resultant graph obtained from the split recomposition of two prn-irreducible graphs, the authors contribute to our understanding of the complexity and representation requirements for such compositions. This knowledge can aid researchers and practitioners in studying and manipulating prn-irreducible graphs in various domains, including network analysis, social sciences, and optimization problems.

In conclusion, this work significantly advances our understanding of word-representable graphs, parity graphs, comparability graphs, and their compositional behaviors. The introduction of permutation representation numbers (prn) and prn-irreducible graphs provides valuable insights into the complexity, structure, and representation requirements of these graph classes. The findings presented in this work pave the way for future research and applications in diverse fields such as computer science, mathematics, network analysis, and optimization.