Word-representability of co-bipartite graph

TL;DR

This work investigates word-representability of co-bipartite graphs by (i) analyzing when complements of bipartite graphs preserve word-representability, (ii) exploring word-representability of co-bipartite graphs with fixed clique sizes, and (iii) characterizing semi-transitive orientations that correspond to word-representable co-bipartite graphs. It shows that the complements of several key bipartite graph families, including path, even cycle, and generalized crown graphs, are word-representable, and it constructs explicit representing words. For fixed clique sizes, the study identifies word-representable structures for clique sizes 2 and 3 under certain neighbor configurations. Finally, it establishes necessary and sufficient conditions for semi-transitive orientations in co-bipartite graphs, tying structural graph properties to the word-representability criterion and offering a framework for further characterization.

Abstract

A graph $G = (V, E)$ is word-representable, if there exists a word $w$ over the alphabet $V$ such that for letters $\{x,y\}\in V$, $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is co-bipartite if its complement is a bipartite graph. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint cliques. The concept of word-representability for co-bipartite graphs has not yet been fully studied. In the book Words and Graphs written by Sergey Kitaev and Vadim Lozin, examples of co-bipartite graphs that are not word-representable are provided. The authors have stated that it remains an open problem to characterize word-representable co-bipartite graphs. It is known that taking the complement of word-representable graphs does not preserve their word-representability. In this paper, we first identify certain classes of bipartite graphs for which word-representation is preserved after the complement operation. We found that the complement of the path graphs, even cycle graphs and generalized crown graphs are also word-representable. Next, we aim to find word-representable co-bipartite graphs in which the size of one clique partition is fixed while the other one can vary. We studied the word-representability of co-bipartite graphs where the sizes of one clique partition are $2$ and $3$. We found that any co-bipartite graphs where the size of the one clique partition is $2$ are word-representable. Also, when the size of the one clique partition is $3$, we found certain co-bipartite graphs are word-representable. Additionally, for word-representable graphs, it has been established that a graph is word-representable if and only if it can be oriented in a specific manner, known as semi-transitive orientation. We provide the necessary and sufficient conditions for a co-bipartite graph to have a semi-transitive orientation.

Figures (21)

• Figure 1: The minimal (by the number of vertices) non-word-representable co-bipartite graphs $\overline{T_1}$
• Figure 2: The minimal (by the number of vertices) non-word-representable co-bipartite graphs $\overline{T_2}$
• Figure 3: Path graph on $6$ vertices
• Figure 4: Cycle graph on $6$ vertices
• Figure 5: induced subgraph of the vertices, $1$, $2$, $3$, $1'$, $2'$ and $3'$ from the graph $\overline{H_{n,n}}$

Theorems & Definitions (52)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Proposition 1.1
• Proposition 1.2
• Example 1.1
• Definition 1.5
• Example 1.2
• Definition 1.6: halldorsson2016semi, Definition 1