Strongly chordal digraphs and Γ\GammaΓ-free matrices (original) (raw)
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List matrix partitions of chordal graphs
Theoretical Computer Science, 2005
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 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 cliques. In an earlier paper we proved that a chordal graph can be partitioned into k independent sets and cliques if and only if it does not contain an induced disjoint union of + 1 cliques of size k + 1. (A linear time algorithm for finding such partitions can be derived from the proof.)
Strong cocomparability graphs and Slash-free orderings of matrices
arXiv (Cornell University), 2022
We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01, 10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time recognition algorithm, for the class. These results complete the picture in which in addition to, or instead of, the Slash matrix one forbids the Γ matrix (which has rows 11, 10). It is well known that in these two cases one obtains the class of interval graphs, and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0, 1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?
Obstructions to partitions of chordal graphs
Discrete Mathematics, 2013
Matrix partition problems generalize graph colouring and homomorphism problems, and occur frequently in the study of perfect graphs. It is difficult to decide, even for a small matrix M , whether the Mpartition problem is polynomial time solvable or NP-complete (or possibly neither), and whether M-partitionable graphs can be characterized by a finite set of forbidden induced subgraphs (or perhaps by some other first order condition). We discuss these problems for the class of chordal graphs. In particular, we classify all small matrices M according to whether M-partitionable graphs have finitely or infinitely many minimal chordal obstructions (for all matrices of size less than four), and whether they admit a polynomial time recognition algorithm or are NP-complete (for all matrices of size less than five). We also suggest questions about larger matrices.
Strongly orderable graphs A common generalization of strongly chordal and chordal bipartite graphs
Discrete Applied Mathematics, 2000
In this paper those graphs are studied for which a so-called strong ordering of the vertex set exists. This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way. We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and prove that these graphs can be recognized in O(|V | + |E|)|V | time. Moreover, a special strong ordering of a strongly orderable graph can be produced in the same time bound. We present variations of greedy algorithms that compute a minimum coloring, a maximum clique, a minimum clique partition and a maximum independent set of a strongly orderable graph in linear time if such a special strong ordering is given.
On the characterization of digraphs with given rank
Linear Algebra and its Applications, 2020
The rank of a digraph is defined as the rank of its adjacency matrix. In this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2k. In particular, since the bipartite graphs of rank 2, 4 and 6 are known, then we characterize digraphs of rank 1, 2 and 3. The results are based on rank-preserving operations on digraphs such as splitting and fusion of vertices or twin-deletion and vertex-multiplication.
NP-completeness results for some problems on subclasses of bipartite and chordal graphs
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 (2003) 2147-2155], 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.
Minimal digraph obstructions for small matrices
arXiv (Cornell University), 2016
Given a {0, 1, * }-matrix M , a minimal M-obstruction is a digraph D such that D is not M-partitionable, but every proper induced subdigraph of D is. In this note we present a list of all the M-obstructions for every 2 × 2 matrix M. Notice that this note will be part of a larger paper, but we are archiving it now so we can cite the results.
Discret. Math. Theor. Comput. Sci., 2018
A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is overrightarrowC_3\overrightarrow{C}_3overrightarrowC3 and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric 333-quasi-transitive and asymmetric 333-anti-quasi-transitive TT3TT_3TT_3-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.
Restricted unimodular chordal graphs
Journal of Graph Theory, 1999
A chordal graph is called restricted unimodular if each cycle of its vertex-clique incidence bipartite graph has length divisible by 4. We characterize these graphs within all chordal graphs by forbidden induced subgraphs, by minimal relative separators, and in other ways. We show how to construct them by starting from block graphs and multiplying vertices subject to a certain restriction, which leads to a linear-time recognition algorithm. We show how they are related to other classes such as distance-hereditary chordal graphs and strongly chordal graphs.