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Electron-phonon interaction and related self-energy are fundamental to boththe equilibrium properties and non-equilibrium relaxation dynamics of solids.Although electron-phonon interaction has been suggested by varioustime-resolved measurements to be important for the relaxation dynamics ofgraphene, the lack of energy- and momentum-resolved self-energy dynamicsprohibits direct identification of the role of specific phonon modes in therelaxation dynamics. Here by performing time- and angle-resolved photoemissionspectroscopy measurements on a Kekul\'e-ordered graphene with folded Diraccones at the $\Gamma$ point, we have succeeded in resolving the self-energyeffect induced by coupling of electrons to two phonons at $\Omega_1$ = 177 meVand $\Omega_2$ = 54 meV and revealing its dynamical change in the time domain.Moreover, these strongly coupled phonons define energy thresholds, whichseparate the hierarchical relaxation dynamics from ultrafast, fast to slow,thereby providing direct experimental evidence for the dominant role ofmode-specific phonons in the relaxation dynamics

Self-energy dynamics and mode-specific phonon threshold effect in a Kekul\'e-ordered graphene