# On forbidden induced subgraphs for K_{1,3}-free perfect graphs

## TL;DR

This paper characterizes certain classes of forbidden induced subgraph graphs, showing conditions under which they are perfect or $ ext{omega}$-colourable, extending previous results on $K_{1,3}$-free graphs.

## Contribution

It provides a complete characterization of $(K_{1,3}, Y)$-free graphs that are perfect or $ ext{omega}$-colourable, identifying specific subgraphs $P_5$, $Z_2$, $P_4$, and $Z_1$ as key.

## Key findings

- Connected $(K_{1,3}, Y)$-free graphs (not odd cycles, independence ≥3) are perfect iff $Y$ is an induced subgraph of $P_5$ or $Z_2$.
- For all connected $(K_{1,3}, Y)$-free graphs, similar characterizations involve $P_4$ and $Z_1$.
- The results extend Chudnovsky and Seymour's work on $K_{1,3}$-free graphs.

## Abstract

Considering connected $K_{1,3}$-free graphs with independence number at least $3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is $2\omega$-colourable where $\omega$ denotes the clique number of $G$. We study $(K_{1,3}, Y)$-free graphs, and show that the following three statements are equivalent. (1) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is perfect. (2) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is $\omega$-colourable. (3) $Y$ is isomorphic to an induced subgraph of $P_5$ or $Z_2$ (where $Z_2$ is also known as hammer). Furthermore, for connected $(K_{1,3}, Y)$-free graphs (without an assumption on the independence number), we show a similar characterisation featuring the graphs $P_4$ and $Z_1$ (where $Z_1$ is also known as paw).

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Source: https://tomesphere.com/paper/1903.09403