7 research outputs found
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
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
Subword complexes, cluster complexes, and generalized multi-associahedra
In this paper, we use subword complexes to provide a uniform approach to
finite type cluster complexes and multi-associahedra. We introduce, for any
finite Coxeter group and any nonnegative integer k, a spherical subword complex
called multi-cluster complex. For k=1, we show that this subword complex is
isomorphic to the cluster complex of the given type. We show that multi-cluster
complexes of types A and B coincide with known simplicial complexes, namely
with the simplicial complexes of multi-triangulations and centrally symmetric
multi-triangulations respectively. Furthermore, we show that the multi-cluster
complex is universal in the sense that every spherical subword complex can be
realized as a link of a face of the multi-cluster complex.Comment: 26 pages, 3 Tables, 2 Figures; final versio
Dynamic Programming for -minor-free Graphs
International audienceWe provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2 O(k*logk)*n O(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for H-minor-free graphs, called an H-minor-free cut decomposition, and we show that they can be constructed in O h (n 3) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called ℓ-triangulation. Our main result is that when applied on H-minor-free cut decompositions, dynamic programming runs in 2Oh(k)⋅nO(1) steps. This broadens substantially the class of problems that can be solved deterministically in single-exponential time for H-minor-free graphs
The size of 3-compatible, weakly compatible split systems
10.1007/s12190-012-0546-zJournal of Applied Mathematics and Computing401-2249-25