Bounded Clique-Width of ($S_{1,2,2}$,Triangle)-Free Graphs

TL;DR

This paper proves that ($S_{1,2,2}$,triangle)-free graphs have bounded clique-width, resolving an open problem and extending to ($P_1+2P_2$,triangle)-free graphs, with implications for graph class complexity.

Contribution

It establishes bounded clique-width for ($S_{1,2,2}$,triangle)-free graphs, solving an open problem and contributing to the understanding of graph class properties.

Findings

($S_{1,2,2}$,triangle)-free graphs have bounded clique-width

Bounded clique-width extends to ($P_1+2P_2$,triangle)-free graphs

Confirms previous partial results and resolves open questions

Abstract

If a graph has no induced subgraph isomorphic to $H_1$ or $H_2$ then it is said to be ($H_1,H_2$)-free. Dabrowski and Paulusma found 13 open cases for the question whether the clique-width of ($H_1,H_2$)-free graphs is bounded. One of them is the class of ($S_{1,2,2}$,triangle)-free graphs. In this paper we show that these graphs have bounded clique-width. Thus, also ($P_1+2P_2$,triangle)-free graphs have bounded clique-width which solves another open problem of Dabrowski and Paulusma. Meanwhile we were informed by Paulusma that in December 2015, Dabrowski, Dross and Paulusma showed that ($S_{1,2,2}$,triangle)-free graphs (and some other graph classes) have bounded clique-width.