of edges and vertices (i.e., the property should hold for all subgraphs). Preserved by the addition of edges and vertices (i.e., the property should hold for all supergraphs). Preserved by the addition of edges. (This meaning is used in the statement of the Aanderaa–Karp–Rosenberg conjecture.) The complementary property of a property that is preserved by the removal of edges is preserved under the addition of edges. Hence some authors avoid this ambiguity by saying a property A is monotone if A or AC (the complement of A) is monotone. Some authors choose to resolve this by using the term increasing monotone for properties preserved under the addition of some object, and decreasing monotone for those preserved under the removal of the same object. In problem solving In planning and problem solving, or more formally one-person games, the search space is seen as a directed graph with states as nodes, and transitions as edges. States can have properties, and such a property P is hereditary if for each state S that has P, each state that can be reached from S also has P. The subset of all states that have P plus the subset of all states that have ~P form a partition of the set of states called a hereditary partition. This notion can trivially be extended to more discriminating partitions by instead of properties, considering aspects of states and their domains. If states have an aspect A, with di ⊂ D a partition of the domain D of A, then the subsets of states for which A ∈ di form a hereditary partition of the total set of states iff ∀i, from any state where A ∈ di only other states where A ∈ di can be reached. If the current state and the goal state are in different elements of a hereditary partition, there is no path from the current state to the goal state — the problem has no solution. Can a checkers board be covered with domino tiles, each of which covers exactly two adjacent fields? Yes. What if we remove the top left and the bottom right field? Then no covering is possible any more, because the difference between number of uncovered white fields and the number of uncovered black fields is 2, and adding a domino tile (which covers one white and one black field) keeps that number at 2. For a total