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.
39190998
Computing Maximum Valued Regions
11379371
Partitioning Graphs into Two Trees
17103586
The complexity of coloring graphs without long induced paths
We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a Ps-free graph is 5-colorable and of deciding whether a Pi2-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a Ps-free graph is 3-colorable.
6054148
Hamiltonian cycles in circulant digraphs with two stripes
Abstract The circulant traveling salesman problem ( CTSP ) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unknown. This paper provides a characterization of Hamiltonian digraphs with circulant distance matrix containing only two nonzero stripes . The corresponding conditions can be checked in polynomial time. Secondly, we show that all Hamiltonian cycles of a circulant 2-digraph are periodic. Based on these two results, a method for enumerating all Hamiltonian cycles in such digraphs is described. Moreover, two simple algorithms are derived for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two nonzero stripes.
431690
Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult
AbstractWe consider the problem of coloring a planar graph with the
minimum number of colors so that each color class avoids one or
more forbidden graphs as subgraphs. We perform a detailed study of
the computational complexity of this problem. We present a complete picture for the case with a single forbidden
connected (induced or noninduced) subgraph. The 2-coloring
problem is NP-hard if the forbidden subgraph is a tree with at
least two edges, and it is polynomially solvable in all other
cases. The 3-coloring problem is NP-hard if the forbidden
subgraph is a path with at least one edge, and it is polynomially solvable in all other
cases. We also derive results for several forbidden sets of
cycles. In particular, we prove that it is NP-complete to decide if a planar
graph can be 2-colored so that no cycle of length at most 5 is
monochromatic.
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)).
5145089
A faster off-line algorithm for the TCP acknowledgement problem
In a recent paper [Proceedings of STOC'98, 1998, pp. 389-398], Dooly, Goldman and Scott study a problem that is motivated by the networking problem of dynamically adjusting delays of acknowledgements in the Transmission Control Protocol (TCP). Among other results, they give an O(n2) off-line algorithm for computing the optimal way of acknowledging n packet arrivals and departures.In this brief note, we observe that there is a faster off-line algorithm for this problem with time complexity O(n).