TOKYO (AP) _ Prosecutors arrested a second former government vice minister today in a widening stock-profiteering scandal that has shaken public confidence in Prime Minister Noboru Takeshita's administration.

The arrest of Kunio Takaishi, 59, who retired last year as vice minister of education, came the same day that formal charges were filed in the case against a former vice minister of labor, Takashi Kato.

Both men, the highest-ranking public officials arrested in the scandal, were accused of receiving bribes in the form of bargain-priced stock shares in exchange for using their influence in favor of Recruit Co., an information conglomerate. Both retired before the scandal surfaced last summer.

Opposition parties have demanded that Takeshita resign and call new elections to allow for a public judgment of the Recruit scandal, which has raised questions about money links between business and government.

Three members of Takeshita's Cabinet have stepped down as a result of the scandal, but the prime minister has vowed to stay on and work for reforms to restore public trust in government. Recent public opinion polls have shown support for the Takeshita government at only 12.6 percent.

Takaishi is accused of helping Recruit executives win appointments to government education committees.

Recruit's main business is publishing specialized magazines on job placement, giving it a strong interest in both education and labor policy.

Kato, 59, who retired as a vice minister in September 1987, was arrested March 8 and accused of influencing a ministry decision not to impose stricter regulations on employment information magazines in 1984.

The latest arrest, the 13th in the Recruit scandal, came a day after prosecutors filed formal charges accusing Hisashi Shinto, former chairman of the telecommunications giant Nippon Telegraph and Telephone Corp., of receiving bribes from Recruit to help the company expand its telecommunicatio ns business.

Prosecutors say Kato was sold 3,000 unlisted shares of Recruit's real estate subsidiary, Recruit-Cosmos Co., at a cut-rate price in 1986. He allegedly profited by about $45,450 when public sales of the stock began and its price soared.

Takaishi bought 10,000 shares and sold 6,000 immediately after the public listing for a profit of about $90,000, prosecutors said.

In all, about 700,000 bargain shares in Recruit-Cosmos reportedly were offered by the founder and former chairman of Recruit, Ezoe Hiromasa, and his aides to more than 150 influential people.

Today, prosecutors added alleged bribery of Kato to the charges already filed against Ezoe and two aides, Masao Tatsumi and Toshihiro Ono. They also arrested Ezoe again and another aide, Hiroshi Kobayashi, in connection with the Takaishi case.

At least 17 leading politicians, including Takeshita and Yasuhiro Nakasone, have been linked to the scandal. Nakasone was prime minister during the stock transactions. However, none have been arrested.