A Refined Characteristic Of Minimum Contingency Set For Conjunctive Query

Given a database instance d, a self join free conjunctive query q and its result q(d), contingency set Gamma(q, d) is a set of tuples from d such that q(d \ Gamma) is false but q(d) is true initially. Finding minimum contingency set Gamma(min)(q, d) is an important problem in database area. An important dichotomy for this problem was identified in the most recent result, Freire et al. showed that Gamma(min)(q boolean AND, d) is NP-complete if the input query includes a triad of form "R-xy, S-yz, T-zx" in a particular manner, PTime otherwise. However, we have two observations: (a) if two clauses have a common variable, then this database instance should be too complex, formally speaking, the visualization of its query result will not be of planarity, this requirement is too strict, (b) there is no limitation on the length of every circle in the visualization of the query result. This makes the previous theorem of dichotomy too weak. Therefore, the natural question is that, if the query result of input database is not of planarity or there is a fixed limitation on the length of every circle, is it Gamma(min)(q Delta, d) still NP-complete? To this end, we strengthen the hardness result, that Gamma(min)(q Delta, d) is still NP-complete, if the input database instance is of planarity, or the maximum length of every circle is limited. Our theorems also generalize the result of triangle edge deletion problem defined on general graph into directed graph, make a contribution to graph theory.