Daniel Meister

Recognition and computation of minimal triangulations for AT-free claw-free and co-comparability graphs

A chordal graph H is a triangulation of a graph G if H is obtained by adding edges to G. If no proper subgraph of H is a triangulation of G we call H a minimal triangulation of G. We introduce a new LexBFS-like breadth-first-search algorithm min-LexBFS. We show that variants of min-LexBFS yield linear-time algorithms for computing minimal triangulations of AT-free claw-free graphs and co-comparability graphs. These triangulation algorithms are used to improve approximation algorithms for the bandwidth of AT-free claw-free and co-comparability graphs. We present a certifying recognition algorithm for proper interval graphs.

Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs ☆

Abstract We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.

Graphs of linear clique-width at most 3

Abstract A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present new characterisations of graphs of linear clique-width at most 2. We also give a layout characterisation of graphs of bounded linear clique-width; a similar characterisation was independently shown by Gurski and by Lozin and Rautenbach.

A new representation of proper interval graphs with an application to clique-width☆

Abstract We introduce a new representation of proper interval graphs that can be computed in linear time and stored in O ( n ) space. This representation is a 2-dimensional vertex partition. It is particularly interesting with respect to clique-width. Based on this representation, we prove new upper bounds on the clique-width of proper interval graphs.

Polar Permutation Graphs

Polar graphs generalise bipartite, cobipartite, split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.

Computing the Clique-width of large path powers in linear time via a new characterisation of clique-width

Clique-width is one of the most important graph parameters, as many NP-hard graph problems are solvable in linear time on graphs of bounded clique-width. Unfortunately, the computation of clique-width is among the hardest problems. In fact, we do not know of any other algorithm than brute force for the exact computation of clique-width on any large graph class of unbounded clique-width. Another difficulty about clique-width is the lack of alternative characterisations of it that might help in coping with its hardness. In this paper, we present two results. The first is a new characterisation of clique-width based on rooted binary trees, completely without the use of labelled graphs. Our second result is the exact computation of the clique-width of large path powers in polynomial time, which has been an open problem for a decade. The presented new characterisation is used to achieve this latter result. With our result, large k-path powers constitute the first non-trivial infinite class of graphs of unbounded clique-width whose clique-width can be computed exactly in polynomial time.

A Complete Characterisation of the Linear Clique-Width of Path Powers

A k -path power is the k -power graph of a simple path of arbitrary length. Path powers form a non-trivial subclass of proper interval graphs. Their clique-width is not bounded by a constant, and no polynomial-time algorithm is known for computing their clique-width or linear clique-width. We show that k -path powers above a certain size have linear clique-width exactly k + 2, providing the first complete characterisation of the linear clique-width of a graph class of unbounded clique-width. Our characterisation results in a simple linear-time algorithm for computing the linear clique-width of all path powers.

Computing treewidth and minimum fill-in for permutation graphs in linear time

A chordal graph H is a triangulation of a graph G, if H is obtained by adding edges to G. If no proper subgraph of H is a triangulation of G, then H is a minimal triangulation of G. A potential maximal clique of G is a set of vertices that induces a maximal clique in a minimal triangulation of G. We will characterise the potential maximal cliques of permutation graphs and give a characterisation of minimal triangulations of permutation graphs in terms of sets of potential maximal cliques. This results in linear-time algorithms for computing treewidth and minimum fill-in for permutation graphs.

A characterisation of clique-width through nested partitions

Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, preserve the tree structures and that they are computable in polynomial time. We apply our characterisation to linear clique-width. And we relate our characterisation to a clique-width characterisation by Heule and Szeider that is used to reduce the clique-width decision problem to a satisfiability problem.

Induced Subgraph Isomorphism on Interval and Proper Interval Graphs

The Induced Subgraph Isomorphism problem on two input graphs G and H is to decide whether G has an induced subgraph isomorphic to H. Already for the restricted case where H is a complete graph the problem is NP-complete, as it is then equivalent to the Clique problem. In a recent paper [7] Marx and Schlotter show that Induced Subgraph Isomorphism is NP-complete when G and H are restricted to be interval graphs. They also show that the problem is W[1]-hard with this restriction when parametrised by the number of vertices in H. In this paper we show that when G is an interval graph and H is a connected proper interval graph, the problem is solvable in polynomial time. As a more general result, we show that when G is an interval graph and H is an arbitrary proper interval graph, the problem is fixed parameter tractable when parametrised by the number of connected components of H. To complement our results, we prove that the problem remains NP-complete when G and H are both proper interval graphs and H is disconnected.