Word-Representability of Graphs with respect to Split Recomposition
Tithi Dwary, K. V. Krishna
TL;DR
This work investigates how word-representability behaves under split recomposition. It proves that word-representable graphs are closed under this operation and determines the representation number of the recomposed graph, establishing $\mathcal{R}(H)=\max\{\mathcal{R}(G),\mathcal{R}(G')\}$. It then analyzes comparability graphs, giving conditions under which recomposition yields a comparability graph and bounding the permutation-representation number $\mathcal{R}^p$ of the result; it also introduces prn-irreducible graphs and derives how their recompositions affect prn. The results provide a modular framework to study word-representability via split components, with parity graphs as a notable corollary and implications for the broader class of perfect graphs through prime components. Overall, the paper links word-representability, poset dimension, and permutation representations in the context of graph decompositions, offering pathways for further classification and complexity considerations.
Abstract
In this work, we show that the class of word-representable graphs is closed under split recomposition and determine the representation number of the graph obtained by recomposing two word-representable graphs. Accordingly, we show that the class of parity graphs is word-representable. Further, we obtain a characteristic property by which the recomposition of comparability graphs is a comparability graph. Consequently, we also establish the permutation-representation number (prn) of the resulting comparability graph. We also introduce a subclass of comparability graphs, called prn-irreducible graphs. We provide a criterion such that the split recomposition of two prn-irreducible graphs is a comparability graph and determine the prn of the resultant graph.