Bounding the Feedback Vertex Number of Digraphs in Terms of Vertex Degrees

Not only in discrete mathematics, generalizing existing concepts and proofs has always been a guiding theme for research. The great mathematician Henri Poincaré even considered this as the leitmotiv of all mathematics. 1 In particular, many results from graph theory were generalized to weighted graphs, digraphs, or hypergraphs. Sometimes, providing such generalizations is an easy exercise; in other cases, the main difficulty lies in formulating the "right generalization" of the original theorem. An additional obstacle is imposed if the result we intend to generalize allows for several proofs or equivalent reformulations. Then there are many roads to potential generalizations to explore, and selecting the most promising one can be difficult. However, once the proper generalizations of the used notions are found, the more general proof often runs very much along the same lines.

As we shall see, one such example is the Turán bound [17], which gives the number of edges that a graph of order n can have when forbidding k-cliques as subgraphs. It allows for many different proofs and equivalent reformulations, see [1]. A dual version of Turán’s bound, regarding the size of independent sets, was refined by Caro [4] and Wei [18]. Their result has subsequently been generalized, by replacing the independent sets with less restricted induced subgraphs [3], respectively by replacing the concept of a graph with more general notions, namely weighted graphs [16] and hypergraphs [5]. Here we complement these efforts by providing a generalization of the Caro-Wei bound to the case of digraphs. From an algorithmic perspective, the new result gauges a simple greedy heuristic for the minimum directed feedback vertex set problem. In this way, the main result of this paper yields a formalized counterpart to the intuition that the minimum (directed) feedback vertex number of sparse digraphs cannot be “overly large”.

We assume the reader is familiar with basic notions in the theory of digraphs, as contained in textbooks such as [10]. Nevertheless, we briefly recall the most important notions in the following. A digraph D = (V, A) consists of a finite set, referred to as the set V(G) = V of vertices, and of an irreflexive binary relation on V(G), referred to as the set of arcs A(G) = A ⊂ V × V. The cardinality of the vertex set is referred to as the order of D. In the special case where the arc relation of a digraph is symmetric, we also speak of an (undirected) graph. For a vertex v in a digraph D, define its out-neighborhood as N + (v) = {u ∈ V | (u, v) ∈ A, u v}, and its outdegree as d + (v) = |N + (v)|. In-neighborhood and in-degree are defined analogously, and denoted by

We note that our definition of vertex degree agrees (on undirected graphs) with the standard usage of this notion in the theory of undirected graphs, see e.g. [7]. For a subset of vertices U ⊆ V of the digraph D = (V, A), the subdigraph induced by U is the digraph (U, A| U×U ) obtained by reducing the vertex set to U and by restricting the arc set to the relation induced by A on U. If a digraph H can be obtained in this way by appropriate restriction of the vertex set of the digraph D, we say

and all start-vertices v i are distinct. If furthermore w k = v 1 , we speak of a cycle. In particular, notice that each pair of opposite arcs (v, w)(w, v) in a digraph amounts to a cycle. This convention is commonly used in the theory of digraphs, compare [10].

A digraph containing no cycles is called acyclic, or a directed acyclic graph (DAG). For a vertex subset U of a digraph D, if the subdigraph induced by U is acyclic, then we call U an acyclic set. In particular, if D[U] contains no arcs at all, then U is called an independent set. The maximum cardinality among all independent sets in D is called the independence number of D. Turán proved the following bound on the independence number of undirected graphs: Theorem 1. Let D = (V, A) be an undirected graph of order n and of average degree d. Then D contains an independent set of size at least d + 1 -1

• n.

Caro [4], and, independently, Wei [18] proved the following refined bound: Theorem 2. Let D = (V, A) be an undirected graph of order n. Then D contains an independent set of size at least v∈V (d(v) + 1) -1 .

The feedback vertex number τ 0 (D) of D is defined as the minimum cardinality among all feedback vertex sets for D. A simple observation is that for a digraph D of order n, the cardinality of a maximum acyclic set equals nτ 0 .

Quite recently, several new algorithms were devised for exactly solving the minimum directed feedback vertex set problem [6,15]. But all known exact algorithms for this problem share the undesirable feature that their worst-case running time is exponential-in the order n of the input graph, or at least in the size of the feedback vertex number τ 0 . This is not surprising as the problem has been known for a long time to be NP-complete, see [8].

Here, we consider the following sim

…(Full text truncated)…

This content is AI-processed based on ArXiv data.