185 research outputs found
Graph Searching Games and Width Measures for Directed Graphs
In cops and robber games a number of cops tries to capture a robber in
a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width or D-width) corresponding to them. All of them generalise tree width and the game characterising it.
For the DAG-width game we prove that the problem to decide the minimal
number of cops required to capture the robber (which is the same as deciding DAG-width), is PSPACE-complete, in contrast to most other similar games. We also show that the cop-monotonicity cost for directed tree width games cannot be bounded by any function. As a consequence, D-width is not bounded in directed tree width, refuting a conjecture by Safari.
A large number of directed width measures generalising tree width has been proposed in the literature. However, only very little was known about the relation between them, in particular about whether classes of digraphs of bounded width in one measure have bounded width in another. In this paper we establish an almost complete order among the most prominent width measures with respect to mutual boundedness
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
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
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
On Complexity of Minimum Leaf Out-branching Problem
Given a digraph , the Minimum Leaf Out-Branching problem (MinLOB) is the
problem of finding in an out-branching with the minimum possible number of
leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved
that MinLOB is polynomial time solvable for acyclic digraphs which are exactly
the digraphs of directed path-width (DAG-width, directed tree-width,
respectively) 0. We investigate how much one can extend this polynomiality
result. We prove that already for digraphs of directed path-width (directed
tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand,
we show that for digraphs of restricted directed tree-width (directed
path-width, DAG-width, respectively) and a fixed integer , the problem of
checking whether there is an out-branching with at most leaves is
polynomial time solvable
On the Monotonicity of Process Number
International audienceGraph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with the tree-and the path-decomposition of graphs. In order to define de-compositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider a game that has been defined and studied in the context of routing reconfiguration problems in WDM networks. Namely, in the processing game, the fugitive is invisible, arbitrarily fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it can access to a strongly connected component free of searchers. We prove that the processing game is monotone which leads to its equivalence with a new digraph decomposition
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