The short answer is depth-first search with backtracking. You can do breadth-first if you prefer, but your little robot will do a lot more walking back and forth then.

More concretely, assume we're given:

// Moves to the given room, which must have a door between
// it and the current room.
moveTo(room);

// Returns the list of room ids directly reachable from
// the current room.
getDoors();

// Returns true if this room is the exit.
isExit();

To find the exit, we just need:

void escape(int startingRoom) {
  Stack<int> path = new Stack<int>();
  path.push(startingRoom);
}

boolean escape(Stack<int> path) {
  for (int door : getDoors()) {
    // Stop if we've escaped.
    if (isExit()) return true;

    // Don't walk in circles.
    if (path.contains(door)) continue;

    moveTo(door);
    path.push(door);
    if (escape(path)) return true;

    // If we got here, the door didn't lead to an exit. Backtrack.
    path.pop();
    moveTo(path.peek());
  }
}

Call escape() with the ID of the start room and it will move the robot to the exit (by calling moveTo()).

The short answer is depth-first search with backtracking. You can do breadth-first if you prefer, but your little robot will do a lot more walking back and forth then.

More concretely, assume we're given:

// Moves to the given room, which must have a door between
// it and the current room.
moveTo(room);

// Returns the list of room ids directly reachable from
// the current room.
getDoors();

// Returns true if this room is the exit.
isExit();

To find the exit, we just need:

void escape(int startingRoom) {
  Stack<int> path = new Stack<int>();
  path.push(startingRoom);
  escape(path);
}

boolean escape(Stack<int> path) {
  for (int door : getDoors()) {
    // Stop if we've escaped.
    if (isExit()) return true;

    // Don't walk in circles.
    if (path.contains(door)) continue;

    moveTo(door);
    path.push(door);
    if (escape(path)) return true;

    // If we got here, the door didn't lead to an exit. Backtrack.
    path.pop();
    moveTo(path.peek());
  }
}

Call escape() with the ID of the start room and it will move the robot to the exit (by calling moveTo()).