EDITING TO CONNECTED F-DEGREE GRAPH.

In the EDGE EDITING TO CONNECTED f-Degree Graph problem we are given a graph G, an integer fc, and a function f assigning integers to vertices of G. The task is to decide whether there is a connected graph F on the same vertex set as G, such that for every vertex v, its degree in F is f(v), and the number of edges in E(G)ΔE(F), the symmetric difference of E(G) and E(F), is at most k. We show that Edge EDITING TO CONNECTED f-DEGREE GRAPH is fixed- parameter tractable (FPT) by providing an algorithm solving the problem on an n-vertex graph in time2O(k)nO(1). We complement this result by showing that the weighted version of the problem with costs 1 and 0 is W[l]-hard when parameterized by k and the maximum value of f even when the input graph is a tree. Our FPT algorithm is based on a nontrivial combination of color-coding and fast computations of representative families over the direct sum matroid of l-elongation of the co-graphic matroid associated with G and a uniform matroid over the set of nonedges of G. We believe that this combination could be useful in designing parameterized algorithms for other edge editing and connectivity problems. [ABSTRACT FROM AUTHOR]