Complexity of stability

摘要

Graph parameters such as the clique number and the chromatic number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs in terms of their computational complexity. We show for various central graph parameters that deciding the stability of a given graph is complete for Θ2p, a well-known complexity class in the second level of the polynomial hierarchy.