We show that every connected induced subgraph of a graph $G$ is dominated by an induced connected split graph if and only if $G$ is $\cal{C}$-free, where $\cal{C}$ is a set of six graphs which includes $P_7$ and $C_7$, and each containing an induced $P_5$. A similar characterisation is shown for the class of graphs which are dominated by induced complete split graphs. Motivated by these results, we study structural descriptions of some classes of $\cal{C}$-free graphs. In particular, we give structural descriptions for the class of ($P_7$,$C_7$,$C_4$,gem)-free graphs and for the class of ($P_7$,$C_7$,$C_4$,diamond)-free graphs. Using these results, we show that every ($P_7$,$C_7$,$C_4$,gem)-free graph $G$ satisfies $\chi(G) \leq 2\omega(G)-1$, and that every ($P_7$,$C_7$,$C_4$,diamond)-free graph $H$ satisfies $\chi(H) \leq \omega(H)+1$. These two upper bounds are tight for any subgraph of the Petersen graph containing a $C_5$.

中文翻译：

---

一些 (P7,C7)-free 图的结构支配和着色

我们表明，当且仅当 $G$ 是 $\cal{C}$-free，其中 $\cal{C}$ 是一个集合时，图 $G$ 的每个连通诱导子图都由一个诱导连通分裂图支配包含 $P_7$ 和 $C_7$ 的六个图，每个图都包含一个诱导的 $P_5$。对于由诱导完全分裂图主导的图类，显示了类似的特征。受这些结果的启发，我们研究了一些无 $\cal{C}$ 图形的结构描述。特别地，我们给出了 ($P_7$,$C_7$,$C_4$,gem)-free 图类和 ($P_7$,$C_7$,$C_4$,diamond)-类的结构描述免费图表。使用这些结果，我们表明每个 ($P_7$,$C_7$,$C_4$,gem)-free 图 $G$ 满足 $\chi(G) \leq 2\omega(G)-1$，并且每个 ($P_7$,$C_7$,$C_4$,diamond)-free 图 $H$ 满足 $\chi(H) \leq \omega(H)+1$。