The following graph represents the positions at Castle Dragonstone. Each edge indicates that the positions are within sight of each other. This is not transitive; i.e., you can't see all the way along a row or column; you can see only the adjacent positions.
King Aerion is putting dragons in his castle. He has a condition that every position in the above graph has to be either occupied by a dragon or within the sight of a dragon. Find the smallest number of dragons that he would need in order to meet this condition.