Representing split graphs by words

TL;DR

This paper investigates the word-representability of split graphs, establishing new characterizations, including forbidden subgraph conditions for certain clique sizes, and explores how combining such graphs affects their word-representability.

Contribution

It proves that threshold graphs are word-representable, characterizes split graphs with large cliques via forbidden subgraphs, and answers an open question on gluing word-representable graphs.

Findings

Threshold graphs are word-representable.

Characterization of split graphs with clique size 5 using forbidden subgraphs.

Gluing two word-representable graphs can or cannot preserve word-representability.

Abstract

There is a long line of research in the literature dedicated to word-representable graphs, which generalize several important classes of graphs. However, not much is known about word-representability of split graphs, another important class of graphs. In this paper, we show that threshold graphs, a subclass of split graphs, are word-representable. Further, we prove a number of general theorems on word-representable split graphs, and use them to characterize computationally such graphs with cliques of size 5 in terms of 9 forbidden subgraphs, thus extending the known characterization for word-representable split graphs with cliques of size 4. Moreover, we use split graphs, and also provide an alternative solution, to show that gluing two word-representable graphs in any clique of size at least 2 may, or may not, result in a word-representable graph. The two surprisingly simple solutions provided by us answer a question that was open for about ten years.