Is a wife considered divorced, Islamically, by the husband's agreement to end the marital relationship and leave the house, or must he explicitly pronounce the divorce?

If the husband refrains from providing maintenance, his wife is permitted to take from his wealth what suffices her and her child, in a reasonable manner. It is desirable for the husband to assist his wife in household chores, as the Prophet, peace and blessings be upon him, used to do. If the wife is someone who cannot serve herself or is ill, the husband is obligated to provide a servant for her. The husband's statement to his wife, "Let's end our relationship," is not a divorce unless he fulfills his promise and divorces her. It is advisable to seek reconciliation and not resort to divorce except out of necessity.