86 research outputs found
On the Galois Lattice of Bipartite Distance Hereditary Graphs
We give a complete characterization of bipartite graphs hav- ing tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. We show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal time computation of neighborhood intersections and maximal bicliques
MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling
Ideal subarrangements of a Weyl arrangement are proved to be free by the
multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao (2016).
They form a significant class among Weyl subarrangements that are known to be
free so far. The concept of MAT-free arrangements was introduced recently by
Cuntz-M{\"u}cksch (2020) to capture a core of the MAT, which enlarges the ideal
subarrangements from the perspective of freeness. The aim of this paper is to
give a precise characterization of the MAT-freeness in the case of type
Weyl subarrangements (or graphic arrangements). It is known that the ideal and
free graphic arrangements correspond to the unit interval and chordal graphs
respectively. We prove that a graphic arrangement is MAT-free if and only if
the underlying graph is strongly chordal. In particular, it affirmatively
answers a question of Cuntz-M{\"u}cksch that MAT-freeness is closed under
taking localization in the case of graphic arrangements.Comment: 25 page
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Vines and MAT-labeled graphs
The present paper explores a connection between two concepts arising from
different fields of mathematics. The first concept, called vine, is a graphical
model for dependent random variables. This concept first appeared in a work of
Joe (1994), and the formal definition was given later by Cooke (1997). Vines
have nowadays become an active research area whose applications can be found in
probability theory and uncertainty analysis. The second concept, called
MAT-freeness, is a combinatorial property in the theory of freeness of
logarithmic derivation modules of hyperplane arrangements. This concept was
first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards
investigated further by Cuntz-M{\"u}cksch (2020).
In the particular case of graphic arrangements, the last two authors (2023)
recently proved that the MAT-freeness is completely characterized by the
existence of certain edge-labeled graphs, called MAT-labeled graphs. In this
paper, we first introduce a poset characterization of a vine, the so-called
vine. Then we show that, interestingly, there exists an explicit equivalence
between the categories of locally regular vines and MAT-labeled graphs. In
particular, we obtain an equivalence between the categories of regular vines
and MAT-labeled complete graphs.
Several applications will be mentioned to illustrate the interaction between
the two concepts. Notably, we give an affirmative answer to a question of
Cuntz-M{\"u}cksch that MAT-freeness can be characterized by a generalization of
the root poset in the case of graphic arrangements.Comment: 32 pages; refined the definitions of the categories MG and LRV (Def.
6.2 & 6.3), hence improved the main result (Thm. 6.10); the term "vineposet"
is no longer used, instead we distinguish the graphical and poset definitions
of a vin
Dually conformal hypergraphs
Given a hypergraph , the dual hypergraph of is the
hypergraph of all minimal transversals of . The dual hypergraph is
always Sperner, that is, no hyperedge contains another. A special case of
Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to
the families of maximal cliques of graphs. All these notions play an important
role in many fields of mathematics and computer science, including
combinatorics, algebra, database theory, etc. In this paper we study
conformality of dual hypergraphs. While we do not settle the computational
complexity status of recognizing this property, we show that the problem is in
co-NP and can be solved in polynomial time for hypergraphs of bounded
dimension. In the special case of dimension , we reduce the problem to
-Satisfiability. Our approach has an implication in algorithmic graph
theory: we obtain a polynomial-time algorithm for recognizing graphs in which
all minimal transversals of maximal cliques have size at most , for any
fixed
Exploiting graph structures for computational efficiency
Coping with NP-hard graph problems by doing better than simply brute force is a field of significant practical importance, and which have also sparked wide theoretical interest. One route to cope with such hard graph problems is to exploit structures which can possibly be found in the input data or in the witness for a solution. In the framework of parameterized complexity, we attempt to quantify such structures by defining numbers which describe "how structured" the graph is. We then do a fine-grained classification of its computational complexity, where not only the input size, but also the structural measure in question come in to play. There is a number of structural measures called width parameters, which includes treewidth, clique-width, and mim-width. These width parameters can be compared by how many classes of graphs that have bounded width. In general there is a tradeoff; if more graph classes have bounded width, then fewer problems can be efficiently solved with the aid of a small width; and if a width is bounded for only a few graph classes, then it is easier to design algorithms which exploit the structure described by the width parameter. For each of the mentioned width parameters, there are known meta-theorems describing algorithmic results for a wide array of graph problems. Hence, showing that decompositions with bounded width can be found for a certain graph class yields algorithmic results for the given class. In the current thesis, we show that several graph classes have bounded width measures, which thus gives algorithmic consequences. Algorithms which are FPT or XP parameterized by width parameters are exploiting structure of the input graph. However, it is also possible to exploit structures that are required of a witness to the solution. We use this perspective to give a handful of polynomial-time algorithms for NP-hard problems whenever the witness belongs to certain graph classes. It is also possible to combine structures of the input graph with structures of the solution witnesses in order to obtain parameterized algorithms, when each structure individually is provably insufficient to provide so under standard complexity assumptions. We give an example of this in the final chapter of the thesis
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