23 research outputs found
A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs
We study the partial search order problem (PSOP) proposed recently by
Scheffler [WG 2022]. Given a graph together with a partial order over the
vertices of , this problem determines if there is an -ordering
that is consistent with the given partial order, where is a graph
search paradigm like BFS, DFS, etc. This problem naturally generalizes the
end-vertex problem which has received much attention over the past few years.
It also generalizes the so-called -tree recognition problem
which has just been studied in the literature recently. Our main contribution
is a polynomial-time dynamic programming algorithm for the PSOP on chordal
graphs with respect to the maximum cardinality search (MCS). This resolves one
of the most intriguing open questions left in the work of Sheffler [WG 2022].
To obtain our result, we propose the notion of layer structure and study
numerous related structural properties which might be of independent interest.Comment: 12 page
Related Orderings of AT-Free Graphs
An ordering of a graph G is a bijection of V(G) to {1, . . . , |V(G)|}. In this thesis, we consider the complexity of two types of ordering problems. The first type of problem we consider aims at minimizing objective functions related to an ordering of the graph. We consider the problems Cutwidth, Imbalance, and Optimal Linear Arrangement. We also consider a problem of another type: S-End-Vertex, where S is one of the following search algorithms: breadth-first search (BFS), lexicographic breadth-first search (LBFS), depth-first search (DFS), and maximal neighbourhood search (MNS). This problem asks if a specified vertex can be the last vertex in an ordering generated by S. We show that, for each type of problem, orderings for one problem may be related to orderings for another problem of that type.
We show that there is always a cutwidth-minimal ordering where equivalence classes of true twins are grouped for any graph, where true twins are vertices with the same closed neighbourhood. This enables a fixed-parameter tractable (FPT) algorithm for Cutwidth on graphs parameterized by the edge clique cover number of the graph and a new parameter, the restricted twin cover number of the graph. The restricted twin cover number of the graph generalizes the vertex cover number of a graph, and is the smallest value k â„ 0 such that there is a twin cover of the graph T and kâ|T| non-trivial components of GâT.
We show that there is also always an imbalance-minimal ordering where equivalence classes of true twins are grouped for any graph. We show a polynomial time algorithm for this problem on superfragile graphs and subsets of proper interval graphs, both subsets of AT-free graphs. An asteroidal triple (AT) is a triple of independent vertices x, y, z such that between every pair of vertices in the triple, there is a path that does not intersect the closed neighbourhood of the third. A graph without an asteroidal triple is said to be AT-free. We also provide closed formulas for Imbalance on some small graph classes.
In the FPT setting, we improve algorithms for Imbalance parameterized by the vertex cover number of the input graph and show that the problem does not have a polynomially sized kernel for the same parameter number unless NP â coNP/poly.
We show that Optimal Linear Arrangement also has a polynomial algorithm for superfragile graphs and an FPT algorithm with respect to the restricted twin cover number.
Finally, we consider S-End-Vertex, for BFS, LBFS, DFS, and MNS. We perform the first systematic study of the problem on bipartite permutation graphs, a subset of AT-free graphs. We show that for BFS and MNS, the problem has a polynomial time solution. We improve previous results for LBFS, obtaining a linear time algorithm. For DFS, we establish a linear time algorithm. All the results follow from the linear structure of bipartite permutation graphs
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
Linear Time LexDFS on Chordal Graphs
Lexicographic Depth First Search (LexDFS) is a special variant of a Depth
First Search (DFS), which was introduced by Corneil and Krueger in 2008. While
this search has been used in various applications, in contrast to other graph
searches, no general linear time implementation is known to date. In 2014,
K\"ohler and Mouatadid achieved linear running time to compute some special
LexDFS orders for cocomparability graphs. In this paper, we present a linear
time implementation of LexDFS for chordal graphs. Our algorithm is able to find
any LexDFS order for this graph class. To the best of our knowledge this is the
first unrestricted linear time implementation of LexDFS on a non-trivial graph
class. In the algorithm we use a search tree computed by Lexicographic Breadth
First Search (LexBFS)
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference âAlgebras, graphs and ordered setâ (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Mauriceâs many scientific interests:âą Lattices and ordered setsâą Combinatorics and graph theoryâą Set theory and theory of relationsâą Universal algebra and multiple valued logicâą Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
Graph Searches and Their End Vertices
Graph search, the process of visiting vertices in a graph in a specific
order, has demonstrated magical powers in many important algorithms. But a
systematic study was only initiated by Corneil et al.~a decade ago, and only by
then we started to realize how little we understand it. Even the apparently
na\"{i}ve question "which vertex can be the last visited by a graph search
algorithm," known as the end vertex problem, turns out to be quite elusive. We
give a full picture of all maximum cardinality searches on chordal graphs,
which implies a polynomial-time algorithm for the end vertex problem of maximum
cardinality search. It is complemented by a proof of NP-completeness of the
same problem on weakly chordal graphs.
We also show linear-time algorithms for deciding end vertices of
breadth-first searches on interval graphs, and end vertices of lexicographic
depth-first searches on chordal graphs. Finally, we present -time algorithms for deciding the end vertices of breadth-first
searches, depth-first searches, maximum cardinality searches, and maximum
neighborhood searches on general graphs
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure