We examine the type-II two-Higgs-doublet model with a light scalar dark matter (S) after imposing the constraints from the Higgs searches at the LHC and dark matter experiments. We first assume that both two CP-even Higgses (h and H) are portals between the DM and SM sectors, and the CP-odd Higgs (A) and H are heavier than 130 GeV. We find that the DM with a mass of 10∼ 50 GeV is disfavored by the joint constraints of the 125 GeV Higgs signal data, the relic density, XENON1T (2017), PandaX-II

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... 017) and the Fermi-LAT. Next, we consider a special scenario in which the heavy CP-even Higgs is taken as the 125 GeV Higgs. The light CP-even Higgs is the only portal between the DM and SM sectors, and the DM mass is slightly below Higgs resonance. We find that the signal data of the 125 GeV Higgs restrict tanβ to be in the range of 1∼ 1.5 for m_h< 62 GeV. The gg→ A→ hZ and bb̅→ h →τ^+τ^- channels at the LHC can impose lower limits and upper limits on tanβ, respectively. For appropriate values of tanβ, λ_h and m_h, the DM with a mass of 10∼ 50 GeV is allowed by the constraints of the Higgs searches at the LHC and dark matter experiments. For example, tanβ is restricted to be in the range of 1.0∼1.5 for 10 GeV 1.125 is excluded for 30 GeV <m_S< 50 GeV.