routing protocol developers

Abstract: Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.

Abstract: A mixed graph $M=(V,E\cup A)$ is acyclic if its directed part $(V,A)$ is an acyclic digraph. In this note we study the so-called orientation completion problem for the class of acyclic mixed graphs. That is, given an acyclic mixed graph $M$ and a property ${\cal P}$; can we orient the edges of $M$ so that the resulting digraph is acyclic and has property ${\cal P}$. We prove that one can decide in polynomial time whether $M$ can be completed to an acyclic digraph with an out-branching from a prescibed vertex $s$, while it is NP-complete to decide whether $M$ has an acyclic orientation which has both an out-branching and an in-branching (a bipolar orientation). We show that it is NP-complete to decide whether $M$ can be oriented so that it contains a directed path between two prescribed vertices. Finally we describe a polynomial algorithm for deciding whether an acyclic digraph $D$ has an out-branching $B^+_s$ such that the digraph $D-A(B^+_s)$ is connected (in the underlying sense). Based on this we pose as an open problem the complexity of deciding whether the edges of an acyclic mixed graph can be oriented so that the result is an acyclic digraph with a non-separating out-branching.