Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Partitions of graphs into one or two independent sets and cliques
Discrete Mathematics
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
Complexity of graph partition problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Linear-time modular decomposition and efficient transitive orientation of comparability graphs
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Finding skew partitions efficiently
Journal of Algorithms
SIAM Journal on Discrete Mathematics
The list partition problem for graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Partitioning chordal graphs into independent sets and cliques
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
Two algorithms for general list matrix partitions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Bi-arc graphs and the complexity of list homomorphisms
Journal of Graph Theory
Digraph matrix partitions and trigraph homomorphisms
Discrete Applied Mathematics
Discrete Applied Mathematics
On injective colourings of chordal graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
On 2-subcolourings of chordal graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
The external constraint 4 nonempty part sandwich problem
Discrete Applied Mathematics
Dichotomy for tree-structured trigraph list homomorphism problems
Discrete Applied Mathematics
Survey: Colouring, constraint satisfaction, and complexity
Computer Science Review
Graph partitions with prescribed patterns
European Journal of Combinatorics
Matrix partitions of split graphs
Discrete Applied Mathematics
Hi-index | 0.00 |
It is well known that a clique with k + 1 vertices is the only minimal obstruction to k-colourability of chordal graphs. A similar result is known for the existence of a cover by l cliques. Both of these problems are in fact partition problems, restricted to chordal graphs. The first seeks partitions into k independent sets, and the second is equivalent to finding partitions into l cliques. In an earlier paper we proved that a chordal graph can be partitioned into k independent sets and l cliques if and only if it does not contain an induced disjoint union of l + 1 cliques of size k + 1. (A linear time algorithm for finding such partitions can be derived from the proof.)In this paper we expand our focus and consider more general partitions of chordal graphs. For each symmetric matrix M over 0, 1, *, the M-partition problem seeks a partition of the input graph into independent sets, cliques, or arbitrary sets, with certain pairs of sets being required to have no edges, or to have all edges joining them, as encoded in the matrix M. Moreover, the vertices of the input chordal graph can be equipped with lists, restricting the parts to which a vertex can be placed. Such (list) partitions generalize (list) colourings and (list) homomorphisms, and arise frequently in the theory of graph perfection. We show that many M-partition problems that are NP-complete in general become solvable in polynomial time for chordal graphs, even in the presence of lists. On the other hand, we show that there are M-partition problems (without lists) that remain NP-complete for chordal graphs. It is not known whether or not each list M-partition problem is NP-complete or polynomial, but it has been shown that each is NP-complete or quasi-polynomial (nO(log n)). For chordal graphs even this 'quasi-dichotomy' is not known, but we do identify large families of matrices M for which dichotomy, or at least quasi-dichotomy, holds.We also discuss forbidden subgraph characterizations of graphs admitting an M-partition. Such characterizations have recently been investigated for partitions of perfect graphs, and we focus on highlighting the improvements one can obtain for the class of chordal, rather than just perfect, graphs.