19 research outputs found
Isometric path complexity of graphs
A set of isometric paths of a graph is "-rooted", where is a
vertex of , if is one of the end-vertices of all the isometric paths in
. The isometric path complexity of a graph , denoted by , is the
minimum integer such that there exists a vertex satisfying the
following property: the vertices of any isometric path of can be
covered by many -rooted isometric paths.
First, we provide an -time algorithm to compute the isometric path
complexity of a graph with vertices and edges. Then we show that the
isometric path complexity remains bounded for graphs in three seemingly
unrelated graph classes, namely, hyperbolic graphs, (theta, prism,
pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively
studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free
graphs are extensively studied in Structural Graph Theory, e.g. in the context
of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied
in Geometric Graph Theory and Computational Geometry. Our results also show
that the distance functions of these (structurally) different graph classes are
more similar than previously thought.
There is a direct algorithmic consequence of having small isometric path
complexity. Specifically, we show that if the isometric path complexity of a
graph is bounded by a constant, then there exists a polynomial-time
constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose
objective is to cover all vertices of a graph with a minimum number of
isometric paths. This applies to all the above graph classes.Comment: A preliminary version appeared in the proceedings of the MFCS 2023
conferenc
Parameterizing Path Partitions
We study the algorithmic complexity of partitioning the vertex set of a given
(di)graph into a small number of paths. The Path Partition problem (PP) has
been studied extensively, as it includes Hamiltonian Path as a special case.
The natural variants where the paths are required to be either \emph{induced}
(Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition,
SPP), have received much less attention. Both problems are known to be
NP-complete on undirected graphs; we strengthen this by showing that they
remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP
remains \NP-hard on undirected bipartite graphs. When parameterized by the
natural parameter ``number of paths'', both SPP and IPP are shown to be
W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected
graphs for the same parameter, as well as for other special subclasses of
directed graphs (IPP is known to be NP-hard on undirected graphs, even for two
paths). On the positive side, we show that for undirected graphs, both problems
are in FPT, parameterized by neighborhood diversity. We also give an explicit
algorithm for the vertex cover parameterization of PP. When considering the
dual parameterization (graph order minus number of paths), all three variants,
IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the
mentioned neighborhood diversity and dual parameterization results to directed
graphs; here, we need to define a proper novel notion of directed neighborhood
diversity. As we also show, most of our results also transfer to the case of
covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of
the CIAC 2023 conferenc
Graph Search Trees and Their Leaves
Graph searches and their respective search trees are widely used in
algorithmic graph theory. The problem whether a given spanning tree can be a
graph search tree has been considered for different searches, graph classes and
search tree paradigms. Similarly, the question whether a particular vertex can
be visited last by some search has been studied extensively in recent years. We
combine these two problems by considering the question whether a vertex can be
a leaf of a graph search tree. We show that for particular search trees,
including DFS trees, this problem is easy if we allow the leaf to be the first
vertex of the search ordering. We contrast this result by showing that the
problem becomes hard for many searches, including DFS and BFS, if we forbid the
leaf to be the first vertex. Additionally, we present several structural and
algorithmic results for search tree leaves of chordal graphs.Comment: full version of an extended abstract to be published in the
Proceedings of the 49th International Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2023) in Fribour
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
Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. However, while every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is motivated by the parallel between moplexes in general graphs and
simplicial modules in chordal graphs: Unlike in the moplex setting, properties
of chordal graphs with a bounded number of simplicial modules are well
understood. For instance, chordal graphs having at most two simplicial modules
are interval. In this work we initiate an investigation of -moplex graphs,
which are defined as graphs containing at most moplexes. Of particular
interest is the smallest nontrivial case , which forms a counterpart to
the class of interval graphs. As our main structural result, we show that the
class of connected -moplex graphs is sandwiched between the classes of
proper interval graphs and cocomparability graphs; moreover, both inclusions
are tight for hereditary classes. From a complexity theoretic viewpoint, this
leads to the natural question of whether the presence of at most two moplexes
guarantees a sufficient amount of structure to efficiently solve problems that
are known to be intractable on cocomparability graphs, but not on proper
interval graphs. We develop new reductions that answer this question negatively
for two prominent problems fitting this profile, namely Graph Isomorphism and
Max-Cut. On the other hand, we prove that every connected -moplex graph
contains a Hamiltonian path, generalising the same property of connected proper
interval graphs. Furthermore, for graphs with a higher number of moplexes, we
lift the previously known result that graphs without asteroidal triples have at
most two moplexes to the more general setting of larger asteroidal sets
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..