Multi-word-representability of graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that for any distinct pair of vertices $x,y\in V$, $(x,y)\in E$ if and only if the occurrences of the letters $x$ and $y$ alternate in the word $w$. Word-representable graphs generalize many important graph classes such as 3-colorable graphs, comparability graphs, and graphs of vertex degree at most 3. The notion of multi-word representability of graphs is a generalization of the notion of word-representability. A graph $G$ is $k$-multi-word-representable if it can be represented by a set of at most $k$ words, each representing a word-representable subgraph of $G$, such that their union is $G$. 2-multi-word-representable graphs contain many graph classes, such as word-representable graphs, planar graphs, interval graphs, split graphs, and line graphs.

We first investigate the problem of finding the largest word-representable induced subgraph of a graph. We solve this problem for graphs of size up to nine. The second problem that we study is determining whether it is possible to vertex partition graphs on $n$ vertices into two sets, each inducing a word-representable graph. We solve this problem for graphs of size up to thirteen and for perfect graphs of size up to sixteen. These results help us to show that graphs with at most 20 vertices are 2-multi-word-representable. The third problem we address involves computing the multi-word-representation number for certain subclasses of perfect graphs. In particular, our investigation reveals that perfect graphs with up to 28 vertices, well-partitioned chordal graphs, and $(6+y)$-trees with at most $(25+y)$ vertices, all subclasses of perfect graphs, are 2-multi-word-representable. Additionally, we establish that any chordal graph is $O(loglogn)$-multi-word-representable.