95 research outputs found
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
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
A tourist guide through treewidth
A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
JGraphT -- A Java library for graph data structures and algorithms
Mathematical software and graph-theoretical algorithmic packages to
efficiently model, analyze and query graphs are crucial in an era where
large-scale spatial, societal and economic network data are abundantly
available. One such package is JGraphT, a programming library which contains
very efficient and generic graph data-structures along with a large collection
of state-of-the-art algorithms. The library is written in Java with stability,
interoperability and performance in mind. A distinctive feature of this library
is the ability to model vertices and edges as arbitrary objects, thereby
permitting natural representations of many common networks including
transportation, social and biological networks. Besides classic graph
algorithms such as shortest-paths and spanning-tree algorithms, the library
contains numerous advanced algorithms: graph and subgraph isomorphism; matching
and flow problems; approximation algorithms for NP-hard problems such as
independent set and TSP; and several more exotic algorithms such as Berge graph
detection. Due to its versatility and generic design, JGraphT is currently used
in large-scale commercial, non-commercial and academic research projects. In
this work we describe in detail the design and underlying structure of the
library, and discuss its most important features and algorithms. A
computational study is conducted to evaluate the performance of JGraphT versus
a number of similar libraries. Experiments on a large number of graphs over a
variety of popular algorithms show that JGraphT is highly competitive with
other established libraries such as NetworkX or the BGL.Comment: Major Revisio
Longest Path and Cycle Transversal and Gallai Families
A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|
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