Minimal feedback vertex sets in directed split‐stars

Abstract

In a graph G = (V , E), a subset F ⊂ V (G) is a feedback vertex set of G if the subgraph induced by V (G)\F is acyclic. In this article, we propose an algorithm for finding minimal feedback vertex sets of directed split-stars. Indeed, our algorithm can derive an upper bound on the size of the minimum feedback vertex set in directed splitstars. Moreover, a simple distributed algorithm is presented for obtaining such sets.

An efficient distributed algorithm for canonical labeling on directed split-stars

“…n is indeed a minimal feedback vertex set of − → S 2 n and an efficient distributed algorithm for exploring the set was presented [14]. The results reveal that the first two leading symbols of each vertex play an important role in determining the characteristics of the vertices in − → S 2 n .…”

“…The directed split-stars are not only strongly connected, but, in fact, they have maximal arc-fault tolerance for connectivity and a small diameter [3][4][5]. Recently, Wang et al developed efficient distributed algorithms to obtain the unique minimum distance-k dominating sets, for k = 1, 2, and minimal feedback vertex sets on n-dimensional directed split-stars [13,14]. This paper proposes a unified approach distributed algorithm to determine membership of the vertex sets with specified leading symbols on directed split-stars.…”

Improved Feedback Vertex Sets in Kautz Digraphs K (d, n)

“…Recently, the lower and the upper bounds to the size of the feedback vertex sets have been established and improved on some graphs, such as hypercubic graphs, meshes, toroids, butterflies, cube-connected cycles, hypercubes and directed split-stars. (see [2], [3], [4], [5], [7], [8], [9], [11], [12], [14], [15]). …”

“…The best known approximation algorithm for this problem has an approximation ratio two [3]. Furthermore, most of the recent researches have been devoted to solving the problem for certain special classes of graphs in polynomial time, such as reducible graphs [14], comparability graphs [10], convex bipartite graphs [10], cyclically reducible graphs [15], and interval graphs [11]. Recently, the lower and the upper bounds to the size of the feedback vertex sets have been established and improved on some graphs, such as hypercubic graphs, meshes, toroids, butterflies, cube-connected cycles, hypercubes and directed split-stars.…”

Feedback Numbers of Goldberg Snark, Twisted Goldberg Snarks and Related Graphs

“…Determining the feedback number is quite difficult even for some elementary graphs. However, the problem has been studied for some special graphs and digraphs, such as hypercubes, meshes, toroids, butterflies, cubeconnected cycles, directed split-stars (see [3,4,5,6,7,8,9,10,11,12,13]). In fact, the minimum feedback set problem is known to be NP-hard for general graphs [14] and the best known approximation algorithm is one with an approximation ratio two [5].…”