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.

翻译：本文证明了词可表示图类在分裂重组下封闭，并确定了两个词可表示图重组所得图的表示数。据此，我们证明了奇偶图是词可表示的。进一步，我们获得了可比较图的重组为可比较图的特征性质。由此我们也建立了最终可比较图的置换表示数（prn）。我们还引入了可比较图的一个子类，称为prn不可约图。我们给出了一个准则，使得两个prn不可约图的分裂重组是可比较图，并确定了所得图的置换表示数。