8 research outputs found
Colorings of oriented planar graphs avoiding a monochromatic subgraph
For a fixed simple digraph and a given simple digraph , an -free
-coloring of is a vertex-coloring in which no induced copy of in
is monochromatic. We study the complexity of deciding for fixed and
whether a given simple digraph admits an -free -coloring. Our main focus
is on the restriction of the problem to planar input digraphs, where it is only
interesting to study the cases . From known results it follows
that for every fixed digraph whose underlying graph is not a forest, every
planar digraph admits an -free -coloring, and that for every fixed
digraph with , every oriented planar graph admits an
-free -coloring.
We show in contrast, that
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring.
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring
The Neighborhood Polynomial of Chordal Graphs
The neighborhood polynomial of a graph is the generating function of
subsets of vertices in that have a common neighbor. In this paper we study
the neighborhood polynomial and the complexity of its computation for chordal
graphs. We will show that it is \NP-hard to compute the neighborhood polynomial
on general chordal graphs. Furthermore we will introduce a parameter for
chordal graphs called anchor width and an algorithm to compute the neighborhood
polynomial which runs in polynomial time if the anchor width is polynomially
bounded. Finally we will show that we can bound the anchor width for chordal
comparability graphs and chordal graphs with bounded leafage. The leafage of a
chordal graphs is the minimum number of leaves in the host tree of a subtree
representation. In particular, interval graphs have leafage at most 2. This
shows that the anchor width of interval graphs is at most quadratic
Well-Separation and Hyperplane Transversals in High Dimensions
A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions.
First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ?^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an ?((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete.
Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ?^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ? d+1 finite point sets in ?^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d