Structural domination and coloring of some (P_7, C_7)-free graphs
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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 C-free, where 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 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 χ(G) ≤ 2ω(G)-1, and that every (P_7,C_7,C_4,diamond)-free graph H satisfies χ(H) ≤ω(H)+1. These two upper bounds are tight for any subgraph of the Petersen graph containing a C_5.
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