26,627 research outputs found
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Algorithms and Complexity Results for Persuasive Argumentation
The study of arguments as abstract entities and their interaction as
introduced by Dung (Artificial Intelligence 177, 1995) has become one of the
most active research branches within Artificial Intelligence and Reasoning. A
main issue for abstract argumentation systems is the selection of acceptable
sets of arguments. Value-based argumentation, as introduced by Bench-Capon (J.
Logic Comput. 13, 2003), extends Dung's framework. It takes into account the
relative strength of arguments with respect to some ranking representing an
audience: an argument is subjectively accepted if it is accepted with respect
to some audience, it is objectively accepted if it is accepted with respect to
all audiences. Deciding whether an argument is subjectively or objectively
accepted, respectively, are computationally intractable problems. In fact, the
problems remain intractable under structural restrictions that render the main
computational problems for non-value-based argumentation systems tractable. In
this paper we identify nontrivial classes of value-based argumentation systems
for which the acceptance problems are polynomial-time tractable. The classes
are defined by means of structural restrictions in terms of the underlying
graphical structure of the value-based system. Furthermore we show that the
acceptance problems are intractable for two classes of value-based systems that
where conjectured to be tractable by Dunne (Artificial Intelligence 171, 2007)
Directed Width Parameters and Circumference of Digraphs
We prove that the directed treewidth, DAG-width and Kelly-width of a digraph
are bounded above by its circumference plus one
Forbidden Directed Minors and Kelly-width
Partial 1-trees are undirected graphs of treewidth at most one. Similarly,
partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known
that an undirected graph is a partial 1-tree if and only if it has no K_3
minor. In this paper, we generalize this characterization to partial 1-DAGs. We
show that partial 1-DAGs are characterized by three forbidden directed minors,
K_3, N_4 and M_5
Directed Minors III. Directed Linked Decompositions
Thomas proved that every undirected graph admits a linked tree decomposition
of width equal to its treewidth. In this paper, we generalize Thomas's theorem
to digraphs. We prove that every digraph G admits a linked directed path
decomposition and a linked DAG decomposition of width equal to its directed
pathwidth and DAG-width respectively
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
- …