Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1217–1227

# Linear Algorithm for a Cyclic Graph Transformation

• V. A. Lyubetsky
• E. Lyubetskaya
• K. Gorbunov
Part 1. Special issue “High Performance Data Intensive Computing” Editors: V. V. Voevodin, A. S. Simonov, and A. V. Lapin

## Abstract

We propose a linear time and linear space algorithm that constructs a minimal (in the total cost) sequence of operations required to transform a directed graph consisting of disjoint cycles into any graph of the same type. The following operations are allowed: double-cut-and-join of vertices and insertion or deletion of a connected fragment of edges; the latter two operations have the same cost. We present a complete proof that the algorithm finds the corresponding minimum. The problem is a nontrivial particular case of the general problem of transforming a graph into another, which in turn is an instance of a hard optimization problem in graphs. In this general problem, which is known to be NP-complete, each vertex of a graph is of degree 1 or 2, edges with the same name may repeat unlimitedly, and operations belong to a well-known list including the above-mentioned operations. We describe our results for the general problem, but the proof is given for the cyclic case only.

## Keywords and phrases

graph cycle graph rearrangement operation cost combinatorial problem optimization in graphs linear algorithm

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## Authors and Affiliations

• V. A. Lyubetsky
• 1
• 2
• E. Lyubetskaya
• 1
• K. Gorbunov
• 1
1. 1.Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)MoscowRussia
2. 2.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia