20 research outputs found
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Efficient Enumeration of Bipartite Subgraphs in Graphs
Subgraph enumeration problems ask to output all subgraphs of an input graph
that belongs to the specified graph class or satisfy the given constraint.
These problems have been widely studied in theoretical computer science. As
far, many efficient enumeration algorithms for the fundamental substructures
such as spanning trees, cycles, and paths, have been developed. This paper
addresses the enumeration problem of bipartite subgraphs. Even though bipartite
graphs are quite fundamental and have numerous applications in both theory and
application, its enumeration algorithms have not been intensively studied, to
the best of our knowledge. We propose the first non-trivial algorithms for
enumerating all bipartite subgraphs in a given graph. As the main results, we
develop two efficient algorithms: the one enumerates all bipartite induced
subgraphs of a graph with degeneracy in time per solution. The other
enumerates all bipartite subgraphs in time per solution
Listing all fixed-length simple cycles in sparse graphs in optimal time
International audienc
Solving maximum clique in sparse graphs: an O ( n m + n 2 d / 4 ) algorithm for d -degenerate graphs
Depth Distribution in High Dimensions
Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set mathcal {B} of axis aligned boxes, whose computation generalizes that of the Kleeâs Measure and of the Maximum Depth. In the worst case over instances of fixed input size n, we describe an algorithm of complexity within (formula presented), using space within mathcal (nlog n), mixing two techniques previously used to compute the Kleeâs Measure. We refine this result and previous results on the Kleeâs Measure and the Maximum Depth for various measures of difficulty of the input, such as the profile of the input and the degeneracy of the intersection graph formed by the boxes
Clause Learning and New Bounds for Graph Coloring
International audienceGraph coloring is a major component of numerous allocation and scheduling problems. We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov's tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree. The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds