Despite all that beckoning wide-open space, the center of the go board is generally not a productive place to try for territory. The real significance of the center is that it is the place of safety. All killing attempts begin with enclosing the target group. When a group runs into the center it becomes virtually impossible to enclose it, and thus the group cant be killed. I realize this may not be very reassuring when your eyeless group is wandering across the board, but you must trust the logic and when your group cant quickly make life on the side you should run it into the center right away.
Why are center groups so hard to kill? The fundamental reason is that each move creates multiple new possibilities. The would-be killer can block one or two of them, but not all. Diagram 1 illustrates the principle (but do not consider it a model for actual play): while Black 1 has four obvious follow-ups -- one-space jumps in four directions -- White can only block one. Black 3 then creates three more follow-ups more one-space jumps, the failsafe center move -- in three directions, and again white can only block one. Black 5 shows another good shape, the dogs neck, and white can only block one of three good follow-up moves. By Black 3 the group is virtually alive, and black has not used any threats to link up to existing stones or to cut off and kill blocking white stones.
All well and good, but youre still worried about safety and eyes. Beginners think that being alive means having eyes, but this is too simple, as well as inaccurate. If a group cant be killed, it is alive, even if it doesnt have eyes. How is this possible? To kill a group, three things must happen: deny it eyes, cut off all connections, and be able to withstand a counter-attack. Without all three of these conditions, a group may be vulnerable, but it cannot definitely be killed. For most of the game, most groups do not have eyes but are not killable. However, the addition of one or more stones at key locations may well turn them into killable groups. One skill in go is to find these key locations in areas near a no-eye but not-killable group which simultaneously threaten the group and create profit in the nearby area. Thinking defensively, one can never assume that a group that the opponent is ignoring can never be attacked. One must keep checking the safety of strong groups, as well as weak groups.
There are two situations when potential center territory becomes significant. The first is in a large structure game when the center potential is added to an existing corner and side structure. The second situation is roughly the second half of a game when the corners and sides are pretty much played out.