اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدC-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width
71
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA95], has only been recently settled [Radoslav Fulek and Csaba D. Toth. Atomic Embeddability, Clustered Planarity, and Thickenability. To appear at SODA20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Toth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.
قيم البحث
اقرأ أيضاً
By a well known result the treewidth of k-outerplanar graphs is at most 3k-1. This paper gives, besides a rigorous proof of this fact, an algorithmic implementation of the proof, i.e. it is shown that, given a k-outerplanar graph G, a tree decomposit
ion of G of width at most 3k-1 can be found in O(kn) time and space. Similarly, a branch decomposition of a k-outerplanar graph of width at most 2k+1 can be also obtained in O(kn) time, the algorithm for which is also analyzed.
In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, textsc{Hamiltonian Cycle} can be solved in time $n^{mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{mathcal{O}(k^2)}$ by Espelage et al
. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $n^{o(k)}$ (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on $k$ vertices with at most $n$ edges.
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchr
onized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of $O(n^{8})$.
A (not necessarily proper) vertex colouring of a graph has clustering $c$ if every monochromatic component has at most $c$ vertices. We prove that planar graphs with maximum degree $Delta$ are 3-colourable with clustering $O(Delta^2)$. The previous b
est bound was $O(Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $Delta$ that exclude a fixed minor are 3-colourable with clustering $O(Delta^5)$. The best previous bound for this result was exponential in $Delta$.
It is known that testing isomorphism of chordal graphs is as hard as the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a tree. The leafage of a chordal graph, is defined to be
the minimum number of leaves in the representing tree. We construct a fixed-parameter tractable algorithm testing isomorphism of chordal graphs with bounded leafage. The key point is a fixed-parameter tractable algorithm finding the automorphism group of a colored order-3 hypergraph with bounded sizes of color classes of vertices.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
