On $(P_5,\bar{P_5})$-sparse graphs and other families

TL;DR

This paper generalizes the concept of sparse graphs by introducing $F$-sparse graphs, providing structural insights into $(P_5,ar{P_5})$-sparse graphs and characterizing those with bounded clique-width.

Contribution

It extends the notion of $P_4$-sparse graphs to $F$-sparse graphs and fully describes the structure of $(P_5,ar{P_5},bull)$-sparse graphs, including their clique-width.

Findings

$(P_5,ar{P_5})$-sparse graphs have known structural properties.

$(P_5,ar{P_5},bull)$-sparse graphs have bounded clique-width.

The paper generalizes sparse graph concepts to finite sets of forbidden subgraphs.

Abstract

We extend the notion of $P_4$-sparse graphs previously introduced by {\scshape Ho\`ang} by considering $\mathcal{F}$-sparse graphs were $\mathcal{F}$ denotes a finite set of graphs on $p$ vertices. Thus we obtain some results on $(P_5,\bar{P_5})$-sparse graphs already known on $(P_5,\bar{P_5})$-free graphs. Finally we completely describe the structure of $(P_5,\bar{P_5}, bull$)-sparse graphs, it follows that those graphs have bounded clique-width.