A program to find families of graphs in Free$\{C_4,4K_1\}$ with bounded clique width

2026-06-15

AI Summary

This study addresses one of the three smallest open cases concerning the complexity of graph coloring under forbidden induced subgraphs of order four, namely whether certain subclasses of the class Free{C₄,4K₁} have bounded clique-width. We introduce a novel (k,l,m)-decomposition framework and prove that any graph class admitting such a decomposition has bounded clique-width. Leveraging this framework, we develop a tailored search procedure to identify several new subclasses with bounded clique-width. Furthermore, we construct an infinite family of 3-clique-covered graphs within Free{C₄,4K₁} whose clique-width is unbounded, thereby revealing the intricate structural complexity of this graph class. Our results establish new connections between graph structure and bounded clique-width, advancing the understanding of this long-standing open problem.

Abstract

In this paper we study the class of graphs without cycles of size 4 and independent sets of size 4 as induced subgraphs: $\mathop{Free}\{C_4, 4K_1\}$. This is one of the three minimal minimal open cases for the complexity of the colouring problem when restricted to classes defined by excluding induced subgraphs of order 4. We investigate the clique width of some subclasses of $\mathop{Free}\{C_4, 4K_1\}$. We introduce a new framework: the $(k,l,m)$-decomposition and prove that if all the graphs of a class $\cal G$ are $(k,l,m)$-decomposable, then graphs in $\cal G$ have bounded clique width. We give a few examples of such class, found with the help of a program we designed. We also show, for any graph $G \in \mathop{Free}\{C_4, 4K_1\}$ that is 3 cliques coverable, an infinite family in $\mathop{Free}\{C_4, 4K_1\}$ of supergraphs of $G$ which have unbounded clique width.

Problem

• clique width
• Free{C4, 4K1}
• induced subgraphs
• graph classes
• bounded clique width

Innovation

• clique width
• (k,l,m)-decomposition
• Free{C4, 4K1}
• graph classes
• induced subgraphs