Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

Graph editing problems have a long history and have been widely studied [35, 53] , with applications in biochemistry [23] and complex network analysis [5]. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property [5, 22]. We consider the problem Graph-Editto-NDL (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. We show Graph-Edit-to-NDL is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter. Golovach and Mertzios [22] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by ∆ + `, where ∆ is the maximum degree of the input graph and ` is the number of edits. We prove Graph-Edit-to-NDL is fixed-parameter tractable when parameterized by ∆ + `. Furthermore, we consider a harder problem Constrained-Graph-Edit-to-NDL (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for Graph-Edit-to-NDL to solve Constrained-Graph-Edit-to-NDL, which proves Constrained-Graph-Editto-NDL is also fixed-parameter tractable when parameterized by ∆ + `. Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by ∆ + `. We show that this family of graph properties includes some well-known graph measures used in complex network analysis.