The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs

AbstractIn this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in “yes”-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W[t] for all t∈N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete

Implementing path coloring algorithms on planar graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures