2 research outputs found
Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm
A circular-arc graph is the intersection graph of arcs of a circle. It is a
well-studied graph model with numerous natural applications. A certifying
algorithm is an algorithm that outputs a certificate, along with its answer (be
it positive or negative), where the certificate can be used to easily justify
the given answer. While the recognition of circular-arc graphs has been known
to be polynomial since the 1980s, no polynomial-time certifying recognition
algorithm is known to date, despite such algorithms being found for many
subclasses of circular-arc graphs. This is largely due to the fact that a
forbidden structure characterization of circular-arc graphs is not known, even
though the problem has been intensely studied since the seminal work of Klee in
the 1960s.
In this contribution, we settle this problem. We present the first forbidden
structure characterization of circular-arc graphs. Our obstruction has the form
of mutually avoiding walks in the graph. It naturally extends a similar
obstruction that characterizes interval graphs. As a consequence, we give the
first polynomial-time certifying algorithm for the recognition of circular-arc
graphs.Comment: 26 pages, 3 figure
A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs
Given a graph , the longest path problem asks to compute a simple path of
with the largest number of vertices. This problem is the most natural
optimization version of the well known and well studied Hamiltonian path
problem, and thus it is NP-hard on general graphs. However, in contrast to the
Hamiltonian path problem, there are only few restricted graph families such as
trees and some small graph classes where polynomial algorithms for the longest
path problem have been found. Recently it has been shown that this problem can
be solved in polynomial time on interval graphs by applying dynamic programming
to a characterizing ordering of the vertices of the given graph
\cite{longest-int-algo}, thus answering an open question. In the present paper,
we provide the first polynomial algorithm for the longest path problem on a
much greater class, namely on cocomparability graphs. Our algorithm uses a
similar - but essentially simpler - dynamic programming approach, which is
applied to a Lexicographic Depth First Search (LDFS) characterizing ordering of
the vertices of a cocomparability graph. Therefore, our results provide
evidence that this general dynamic programming approach can be used in a more
general setting, leading to efficient algorithms for the longest path problem
on greater classes of graphs. LDFS has recently been introduced in
\cite{Corneil-LDFS08}. Since then, a similar phenomenon of extending an
existing interval graph algorithm to cocomparability graphs by using an LDFS
preprocessing step has also been observed for the minimum path cover problem
\cite{Corneil-MPC}. Therefore, more interestingly, our results also provide
evidence that cocomparability graphs present an interval graph structure when
they are considered using an LDFS ordering of their vertices, which may lead to
other new and more efficient combinatorial algorithms.Comment: 24 pages, 4 figures, 4 algorithm