In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $λ_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution $ξ$ has an exponential tail, i.e., $\mathbb{E} e^{cξ}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett [12]. On the random graph with degree distribution $μ$, we show that if $μ$ has an exponential tail, then for small enough $λ$ the contact process with the all-infected initial condition survives for $n^{1+o(1)}$-time w.h.p. (short survival), while for large enough $λ$ it runs over $e^{Θ(n)}$-time w.h.p. (long survival). When $μ$ is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed $λ>0$.