44716494
Polynomial Graph-Colorings
For directed graphs G and H, we say that G is H-colorable, if there is a graph homomorphism from G into H; that is, there is a mapping f from the vertex set of G into the vertex set of H such that whenever there is an edge (x, y) in G, then (f(x), f(y)) is an edge in H. We introduce a new technique for proving that the H-coloring problem is polynomial time decidable for some fixed graphs H. Among others, this is the case if H is a semipath (a “path” with edges directed in either direction), which has not been known before. We also show the existence of a tree T, for which the T-coloring problem is NP-complete.
42221744
Exact algorithms for the Hamiltonian cycle problem in planar graphs
We construct an exact algorithm for the Hamiltonian cycle problem in planar graphs with worst case time complexity O(c^n), where c is some fixed constant that does not depend on the instance. Furthermore, we show that under the exponential time hypothesis, the time complexity cannot be improved to O(c^o^(^n^)).
24777938
More About Subcolorings
A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings.On the negative side, we prove that 2-subcoloring is NP-hard for comparability graphs, and that 3-subcoloring is NP-hard for AT-free graphs and for complements of planar graphs. On the positive side, we derive polynomial time algorithms for 2-subcoloring of complements of planar graphs, and for r-subcoloring of interval and of permutation graphs. Moreover, we prove asymptotically best possible upper bounds on the subchromatic number of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.
25160918
Radio Labeling with Pre-assigned Frequencies
A radio labeling of a graph G is an assignment of pairwise distinct, positive integer labels to the vertices of G such that labels of adjacent vertices differ by at least 2. The radio labeling problem (RL) consists in determining a radio labeling that minimizes the maximum label that is used (the so-called span of the labeling). RL is a well-studied problem, mainly motivated by frequency assignment problems in which transmitters are not allowed to operate on the same frequency channel. We consider the special case where some of the transmitters have preassigned operating frequency channels. This leads to the natural variants P-RL(l) and P-RL(*) of RL with l pre-assigned labels and an arbitrary number of pre-assigned labels, respectively.We establish a number of combinatorial, algorithmical, and complexity-theoretical results for these variants of radio labeling. In particular, we investigate a simple upper bound on the minimum span, yielding a linear time approximation algorithm with a constant additive error bound for P-RL(*) restricted to graphs with girth ? 5. We consider the complexity of P-RL(l) and P-RL(*) for several cases in which RL is known to be polynomially solvable. On the negative side, we prove that P-RL(*) is NP-hard for cographs and for k-colorable graphs where a k-coloring is given (k ? 3). On the positive side, we derive polynomial time algorithms solving P-RL(*) and P-RL(l) for graphs with bounded maximum degree, and for solving P-RL(l) for k-colorable graphs where a k-coloring is given.
38847115
The complexity of detecting crossingfree configurations in the plane
The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.
14343122
Backbone Colorings for Networks
We study backbone colorings, a variation on classical vertex colorings: Given a graph G=(V,E) and a spanning subgraph H (the backbone) of G, a backbone coloring for G and H is a proper vertex coloring V →{ 1,2,... } in which the colors assigned to adjacent vertices in H differ by at least two. We concentrate on the cases where the backbone is either a spanning tree or a spanning path.
43889321
It is tough to be a plumber
In the Linux computer game KPlumber, the objective is to rotate tiles in a raster of squares so as to complete a system of pipes. We give a complexity classification for the original game and various special cases of it that arise from restricting the set of six possible tiles.Most of the cases are NP-complete. One polynomially solvable case is settled by formulating it as a perfect matching problem; other polynomial cases are settled by simple sweepline techniques. Moreover, we show that all the unsettled cases are polynomial time equivalent.
15492927
Counting Convex Polygons in Planar Point Sets
Given a set S of n points in the plane, we compute in time O(n3) the total number of convex polygons whose vertices are a subset of S. We give an O(m · n3) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3,…, m; previously known bounds were exponential (O(nk/2). We also compute the number of empty convex polygons (resp.,k-gons, k m) with vertices in S in time O(n3) (resp., O(m · n3)).