Partitioning Perfect Graphs into Stars

2002

Combinatorica, 2007

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/ √ lg n). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C · cp(G)/ √ lg n classes, then NP ⊆ RTime(n O(lg lg n) ). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lg n) c ) for some constant c strictly between 0 and 1. * The work reported here is a merger of the results reported in [KRS05] and .

The Satisfactory Partition problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler (GK98, GK00) and further studied by other authors but its complexity re- mained open until now. We prove in this paper that Satisfactory Partition, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solv- able. We also study generalizations and variants of this problem where a partition into k nonempty parts (k 3) is requested.

Information Processing Letters, 1988

The Uniform Partition (UP) and the Simple Max Partition (SMP) problems arc NP-complete graph partitioning problems, and polynomial-time algorithms are known for few classes of graphs only. In the present paper, the class of line-graphs is considered and a polynomial algorithm is proposed to solve both problems in this class. When the instance space is extended to digraphs, a characterization is possible which leads to similar results.

The American Mathematical Monthly, 2015

Given a graph G on n vertices, for which m is it possible to partition the edge set of the m-fold complete graph mK n into copies of G? We show that there is an integer m 0 , which we call the partition modulus of G, such that the set M (G) of values of m for which such a partition exists consists of all but finitely many multiples of m 0. Trivial divisibility conditions derived from G give an integer m 1 which divides m 0 ; we call the quotient m 0 /m 1 the partition index of G. It seems that most graphs G have partition index equal to 1, but we give two infinite families of graphs for which this is not true. We also compute M (G) for various graphs, and outline some connections between our problem and the existence of designs of various types.

European Journal of Combinatorics, 2014

We discuss partition problems that generalize graph colouring and homomorphism problems, and occur frequently in the study of perfect graphs. Depending on the pattern, we seek a finite forbidden induced subgraph characterization, or at least a polynomial time algorithm. We give an overview of the current knowledge, focusing on open problems and recent breakthroughs.

Theoretical Computer Science, 2007

Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Comput. Sci. 290 (2003Sci. 290 ( ) 2147Sci. 290 ( -2155]], where he left the problem open for the class of convex graphs, we prove that the kpath partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135-138], we show NPcompleteness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the k-path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the k-path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.

In this paper, we continue the investigation made in about the approximability of P k partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the P k partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum P k packing problem. Moreover, we propose some simple approximation algorithms for those problems.

1992

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by dt(G). We extend considerably the known hardness results by showing it is NP-complete to decide whether dt(G) ≥ 3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k ≥ 3, it is NP-complete to decide whether dt(G) ≥ k, where G is a split graph or k-regular. In particular, these results complement recent combinatorial results regarding dt(G) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2 n n O(1) time, and derive even faster algorithms for special graph classes.