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We show that every connected induced subgraph of a graph $G$ is dominated byan 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 eachcontaining an induced $P_5$. A similar characterisation is shown for the classof graphs which are dominated by induced complete split graphs. Motivated bythese results, we study structural descriptions of some classes of$\cal{C}$-free graphs. In particular, we give structural descriptions for theclass 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 thatevery ($P_7$,$C_7$,$C_4$,gem)-free graph $G$ satisfies $\chi(G) \leq2\omega(G)-1$, 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 anysubgraph of the Petersen graph containing a $C_5$.

Structural domination and coloring of some ($P_7, C_7$)-free graphs