1 research outputs found
Minimum Reload Cost Graph Factors
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 the 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 that of
finding an -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 , 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 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