Abstract
Graphs and Algorithms For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.
Highlights
For a finite undirected graph G = (V, E), the minimum feedback vertex set problem is to find a set of vertices in G of minimum cardinality whose removal induces a forest
We study the minimum feedback vertex set (MFVS) problem on finite distance graphs and circulant graphs
The approximation ratios of our algorithms are in many cases equal or asymptotically close to one for circulant graphs and for distance graphs with chords of length at most half the number of vertices
Summary
For a finite undirected graph G = (V, E), the minimum feedback vertex set problem is to find a set of vertices in G of minimum cardinality whose removal induces a forest. A minimum feedback vertex set (MFVS) is a FVS of cardinality F (G). This parameter has received much attention since finding a MFVS of a graph has a lot of applications in various areas such as combinatorial circuit design, deadlock prevention in computer systems, VLSI testing, artificial intelligence [BG94] and converter placement in optical networks [Tog00]. We study the MFVS problem on finite distance graphs and circulant graphs. The finite distance graph Pn(D) is the subgraph of P (D) induced by vertices. Any FVS must have at least m−n+1 ∆−1 vertices
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