Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

Abstract

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $α_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then $α_{\mathcal{F}}(G) \le (r-1)kγ(G)$, where $γ(G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $α(G) \le 2γ(G)$ for $K_{1,3}$-free (claw-free) graphs, where $α(G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $α(G) \le 2\left(\frac{d+1}{d+2}\right)γ(G)$, which is tight for $d \in \{2, 3, 4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have $α_{\mathcal{F}}(G) \le r(K_r, \mathcal{F^*})γ(G)$, where $\mathcal{F^*}$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show $α_{\mathcal{F}}(G) \le (r(K_q, K_r) - 1)γ(G)$. Finally, for the \emph{$k$-independence number} $α_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $δ\ge k+1$, \[ α_k(G) \le \left( \frac{(r-1)(k+1)}{δ- k + (r-1)(k+1)} \right) n, \] and this bound is sharp for all parameters.