Covering Directed Graphs by In-Trees (Computing and Combinatorics)

Abstract

Given a directed graph D = (V, A) with a set of d specified vertices S = {s 1, ..., s d } ⊆ V and a function where ℤ + denotes the set of non-negative integers, we consider the problem which asks whether there exist in-trees denoted by for every i = 1, ..., d such that are rooted at s i , each T i, j spans vertices from which s i is reachable and the union of all arc sets of T i, j for i = 1, ..., d and j = 1, ..., f(s i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.