Retaining the plasmon Landau damping rate, we analyzed the dispersion relation of the two-plasmon decay (TPD) instability of an electromagnetic wave (pump). Thereby we obtained the threshold pump intensity to get the vertex point of the hyperbola-lemniscate curve as one of the k-matching corners for the maximum growth rate. We call this the ‘vertex threshold.’ The vertex threshold has a significance for the propagation spectrum of electron plasma waves amplified by the TPD instability. Thus, when the pump intensity is below the vertex threshold, there appears a funnel-shaped vacancy of plasmon propagation around the pump-parallel (target-oriented) direction. However, when the pump intensity exceeds the vertex threshold, the whole spectrum from to (with respect to the pump direction) is expected, which was formerly known in the strong-coupling limit of the TPD instability. Due to the effect of the k-dependence of the plasmon Landau damping rate, the vertex threshold increases rapidly with electron temperature. For example, the vertex threshold is at 4.5 keV and is at 13.6 keV, in terms of the electron quiver velocity amplitude divided by the speed of light in a vacuum. So, for a high-temperature plasma, the plasmon vacancy might persist around the target-oriented direction under a rather high pump intensity.