The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to the problem of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of r = 2, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT-algorithm.

充值成本的概念图表中的“经度”是指在通过一个顶点的两个入射边缘遍历顶点时发生的成本。该成本由两个边缘的颜色唯一确定。该概念在运输网络，通信网络和能量分配网络中具有各种应用。在文献中定义并研究了使用该模型的各种问题。找到直径相对于重载成本最小的生成树的问题，找到两个给定顶点之间的总重载成本最小的路径，步道或步行的问题，找到图的适当边缘着色的问题总重载成本被最小化，找到生成树的问题使得所有成对顶点之间的所有路径的重载成本之和被最小化，此类问题的例子包括找到一组覆盖图的所有顶点的最小重装成本循环的问题。在这项工作中，我们专注于最后一个问题。注意图的循环覆盖是它的2因子，因此我们将问题概括为寻找曲线图的问题。边缘彩色图的最小重装成本的r因子。我们针对问题的特殊情况证明了几个NP硬度结果。即，有界度图，平面图，有界总成本和有区别成本的有界数。对于r = 2的特殊情况，我们的结果暗示了改进的NP硬度结果。在积极方面，我们提出了多项式时间可解的特例，该特例在r和图的最大程度方面在多项式和难例之间提供了紧密的边界。然后，我们研究问题的参数化复杂度，证明W [1]-硬度结果并提出FPT算法。