160 research outputs found
Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs
We present space-efficient algorithms for computing cut vertices in a given
graph with vertices and edges in linear time using bits. With the same time and using bits, we can compute the
biconnected components of a graph. We use this result to show an algorithm for
the recognition of (maximal) outerplanar graphs in time using
bits
A characterization of horizontal visibility graphs and combinatorics on words
An Horizontal Visibility Graph (for short, HVG) is defined in association
with an ordered set of non-negative reals. HVGs realize a methodology in the
analysis of time series, their degree distribution being a good discriminator
between randomness and chaos [B. Luque, et al., Phys. Rev. E 80 (2009),
046103]. We prove that a graph is an HVG if and only if outerplanar and has a
Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in
algebraic combinatorics [P. Flajolet and M. Noy, Discrete Math., 204 (1999)
203-229]. Our characterization of HVGs implies a linear time recognition
algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs
highlighting various connections with combinatorial statistics and introducing
the notion of a visible pair. With this technique we determine asymptotically
the average number of edges of HVGs.Comment: 6 page
Maximal outerplanar graphs as chordal graphs, path-neighborhood graphs, and triangle graphs
Maximal outerplanar graphs are characterized using three different classes of graphs. A path-neighborhood graph is a connected graph in which every neighborhood induces a path. The triangle graph has the triangles of the graph as its vertices, two of these being adjacent whenever as triangles in they share an edge. A graph is edge-triangular if every edge is in at least one triangle. The main results can be summarized as follows: the class of maximal outerplanar graphs is precisely the intersection of any of the two following classes: the chordal graphs, the path-neighborhood graphs, the edge-triangular graphs having a tree as triangle graph.maximal outerplanar graph;path-neighborhood graph;triangle graph;chordal graph;elimination ordering
Frequent Subgraph Mining in Outerplanar Graphs
In recent years there has been an increased interest in frequent pattern discovery in large databases of graph structured objects. While the frequent connected subgraph mining problem for tree datasets can be solved in incremental polynomial time, it becomes intractable for arbitrary graph databases. Existing approaches have therefore resorted to various heuristic strategies and restrictions of the search space, but have not identified a practically relevant tractable graph class beyond trees. In this paper, we define the class of so called tenuous outerplanar graphs, a strict generalization of trees, develop a frequent subgraph mining algorithm for tenuous outerplanar graphs that works in incremental polynomial time, and evaluate the algorithm empirically on the NCI molecular graph dataset
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth
We investigate crossing minimization for 1-page and 2-page book drawings. We
show that computing the 1-page crossing number is fixed-parameter tractable
with respect to the number of crossings, that testing 2-page planarity is
fixed-parameter tractable with respect to treewidth, and that computing the
2-page crossing number is fixed-parameter tractable with respect to the sum of
the number of crossings and the treewidth of the input graph. We prove these
results via Courcelle's theorem on the fixed-parameter tractability of
properties expressible in monadic second order logic for graphs of bounded
treewidth.Comment: Graph Drawing 201
Frequent Subgraph Mining in Outerplanar Graphs
In recent years there has been an increased interest in frequent pattern discovery in large databases of graph structured objects. While the frequent connected subgraph mining problem for tree datasets can be solved in incremental polynomial time, it becomes intractable for arbitrary graph databases. Existing approaches have therefore resorted to various heuristic strategies and restrictions of the search space, but have not identified a practically relevant tractable graph class beyond trees. In this paper, we define the class of so called tenuous outerplanar graphs, a strict generalization of trees, develop a frequent subgraph mining algorithm for tenuous outerplanar graphs that works in incremental polynomial time, and evaluate the algorithm empirically on the NCI molecular graph dataset
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
Maximal outerplanar graphs as chordal graphs, path-neighborhood graphs, and triangle graphs
Maximal outerplanar graphs are characterized using three different classes of graphs. A path-neighborhood graph is a connected graph in which every neighborhood induces a path. The triangle graph has the triangles of the graph as its vertices, two of these being adjacent whenever as triangles in they share an edge. A graph is edge-triangular if every edge is in at least one triangle. The main results can be summarized as follows: the class of maximal outerplanar graphs is precisely the intersection of any of the two following classes: the chordal graphs, the path-neighborhood graphs, the edge-triangular graphs having a tree as triangle graph
Generating Outerplanar Graphs Uniformly at Random
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.We show how to generate labelled and unlabelled outerplanar graphs with vertices uniformly at random in polynomial time in . To generate labelled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labelled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs.
Next we modify our formulas to also count rooted unlabelled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabelled outerplanar graphs uniformly at random in expected polynomial time.Peer Reviewe
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