We found a very large, 35 kpc, emission-line region around the Seyfert type 2 galaxy NGC 4388, using deep narrow-band imaging with the prime focus camera (Suprime-Cam) of the Subaru telescope. This region consists of many faint gas clouds or filaments, and extends northeastwards from the galaxy. The typical H-alpha luminosity L(H-alpha) of the clouds is 10^37 erg s^-1, and the total L(H-alpha) of the region within 10 kpc from the nucleus is 2 x 10^38 erg s^-1, which corresponds to an ionized gas mass of 10^5 M_solar. The map of the emission-line intensity ratio I([OIII])/I(H-alpha) indicates that the inner (r < 12 kpc) region of the VEELR may be excited by nuclear ionizing radiation. The excitation mechanism of the outer (r > 12 kpc) region is unclear, but it is likely that the nuclear radiation is also a dominant source of its ionization. We discuss the origin of the ionized gas. Two plausible origins of the gas in the VEELR are (i) the tidal debris resulting from a past interaction with a gas-rich dwarf galaxy, i.e., a minor merger, or (ii) the interstellar medium of NGC 4388, stripped by the ram pressure of the hot intracluster medium of the Virgo cluster.
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