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The feedback vertex number of, denoted by, is the cardinality of a minimum feedback vertex set of.
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A set is called a feedback vertex set of if contains no cycle. Actually, it is shown that, and if then. In this note, we determine and for the Cartesian product of directed cycles. arcs) whose removal leaves the resultant digraph free of directed cycles. the feedback arc number ) is the minimum number of vertices, (resp. (This paper is a revised version of Antoine Bossard (2020)).įor a digraph, the feedback vertex number, (resp. We describe in this paper a decycling algorithm for the 3-dimensional k-ary torus topology and compare it with established results, both theoretically and experimentally. Hence, network decycling is an important issue, and it has been extensively discussed in the literature. It is well known that cycles harm the parallel processing capacity of systems: for instance, deadlocks and starvations are two notorious issues of parallel computing that are directly linked to the presence of cycles. Given the high number of compute nodes in such systems, efficient parallel processing is critical to maximise the computing performance. Notably, this network topology is employed by the supercomputer ranked number one in the world as of November 2020, the supercomputer Fugaku. The torus topology has proven very popular for the interconnect of these high-performance systems. Modern supercomputers are massively parallel systems: they embody thousands of computing nodes and sometimes several millions. This problem does not have a simple solution. The corresponding problem of eliminating all cycles from a graph by means of deletion of vertices goes back at least to the work of Kirchho (16) on spanning trees. Decycling a Graph The minimum number of edges whose removal eliminates all cycles in a given graph has been known as the cycle rank of the graph, and this parameter has a simple expression: b(G) =kGkj Gj +! ((14), Chapter 4) where, as in (12),jGj andkGk are respectively the number of vertices and the number of edges of G and ! is the number of components of G. A structural description of graphs with a xed decycling number based on connectivity is also presented. Results to be reviewed include recent work on decy- cling numbers of cubes, grids and snakes and bounds on the decycling number of cubic graphs, and expected bounds on the decycling numbers of random regular graphs. The purpose of this paper is to provide a review of recent results and open problems on this parameter. The size of a smallest decycling set of G is called the decycling number of G. For a graph G and S V (G), if G S is acyclic, then S is said to be a decycling set of G.