On cleaving a planar graph
shankar venkatesan1990, Discrete Mathematics
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
We first show that the removal of 4fi vertices from an n-vertex planar graph with non-negative vertex weights summing to no more than 1 is sufficient to cleave or recursively separate it into components of weight no more than a given E, thus improving on the 2fia bound shown in . We then derive worst-case bounds on the number of vertices necessary to separate a planar graph of a given radius into components of weight no more than E.
Related papers
Well-Connected Separators for Planar Graphs
2000
Given an n-vertex weighted planar graph G, a separator is a subset S of vertices such that each component of G ? S has at most two-thirds of the original weight. We give an algorithm nding a separator while balancing these two parameters of G S]: the number of components, and the cost of a minimum spanning forest. In particular, given uniform edge costs and a positive integer k p n, we nd a separator S such that jSj = O(n=k) and G S] has O(k) components.
On Edge-Partitioning of Complete Geometric Graphs into Plane Trees
arXiv: Combinatorics, 2019
In response to a well-known open question ``Does every complete geometric graph on $2n\\/$ vertices have a partition of its edge set into $n\\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel configuration. Also we present sufficient conditions for the complete geometric graph on $2n\\/$ vertices to have a partition of its edge set into $n\\/$ plane spanning trees (which are double stars, caterpillars or $ w\\/$-caterpillars).
A Linear-Time Algorithm to Find Independent Spanning Trees in Maximal Planar Graphs
Lecture Notes in Computer Science, 2000
and Shin-ichi NAKANO †a) , Regular Member SUMMARY Given a graph G, a designated vertex r and a natural number k, we wish to find k "independent" spanning trees of G rooted at r, that is, k spanning trees such that, for any vertex v, the k paths connecting r and v in the k trees are internally disjoint in G. In this paper we give a linear-time algorithm to find k independent spanning trees in a k-connected maximal planar graph rooted at any designated vertex.
Degree-constrained decompositions of graphs: Bounded treewidth and planarity
Theoretical Computer Science, 2006
We study the problem of decomposing the vertex set V of a graph into two nonempty parts V 1 , V 2 which induce subgraphs where each vertex v ∈ V 1 has degree at least a(v) inside V 1 and each v ∈ V 2 has degree at least b(v) inside V 2 . We give a polynomialtime algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.
Partitioning planar graphs: a fast combinatorial approach for max-cut
Computational Optimization and Applications, 2012
The max-cut problem asks for partitioning the nodes V of a graph G = (V, E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NPhard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. [45] and that of Berman et al. [9]. The running time of the former can be bounded by O(|V | 3 2 log |V |). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O([V | 3 2 (log |V |) 3 2)α(|V |), where α(|V |) is the inverse Ackermann function, it can solve large instances in practice. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O(|V | 3 2 log |V |), similar to the bound achieved by [45]. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10 6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of [45] and [9]. It turns out that our algorithm is considerably faster in practice than [45]. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of [9]. However, whereas the procedure of generating the expanded graph in [9] is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task.
The Linear 2-Arboricity of Planar Graphs
Graphs and Combinatorics, 2003
Let G be a planar graph with maximum degree D and girth g. The linear 2-arboricity la 2 ðGÞ of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la 2 ðGÞ dðD þ 1Þ=2e þ 12; (2) la 2 ðGÞ dðD þ 1Þ=2e þ 6 if g ! 4; (3) la 2 ðGÞ dðD þ 1Þ=2e þ 2 if g ! 5; (4) la 2 ðGÞ dðD þ 1Þ=2e þ 1 if g ! 7.
The complexity of the matching-cut problem for planar graphs and other graph classes
Journal of Graph Theory, 2009
The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be N P-complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains N P-complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is N P-complete for planar graphs with girth five. The reduction is from planar graph 3-colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching-cuts are described. These classes include claw-free graphs, co-graphs, and graphs A preliminary version of this
Thin Trees in 8-edge-Connected Planar Graphs
Information Processing Letters, 2018
Merker and Postle showed that any graph that can be decomposed into a forest and a star forest has a decomposition into two forests with diameter at most 18. Using this result, they proved any planar graph of girth at least 6 has two edge-disjoint 18 19-thin spanning trees. By using a simpler version of their techniques, given a graph that can be decomposed into a forest and a matching, in polynomial time, we decompose the graph into two forests with diameter at most 12. Using this result, we are able to show that any 8-edge-connected planar graph has two edge-disjoint 12 13-thin spanning trees.
Linear time low tree-width partitions and algorithmic consequences
2006
Abstract Classes of graphs with bounded expansion have been introduced in [15],[12]. They generalize both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N (C, p) so that the vertex set of any graph G∈ C may be partitioned into at most N (C, p) parts, any i≤ p parts of them induce a subgraph of tree-width at most (i-1)[12](actually, of tree-depth [16] at most i, what is sensibly stronger).
A linear-time algorithm for edge-disjoint paths in planar graphs
Combinatorica, 1995
In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the "classical" case where an instance additionally fulfills the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires O (nb/3(loglogn)l/3) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in an O(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.
Related topics
Related papers
Partitions of Graphs into Trees
Lecture Notes in Computer Science, 2007
In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder's realizers , which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NP-hardness of the k-tree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques .
On the vertex-arboricity of planar graphs
European Journal of Combinatorics, 2008
The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G) ≤ 3 for any planar graph G. In this paper we prove that a(G) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.
A note on packing spanning trees in graphs and
2014
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
Matching and l-subgraph contractibility to planar graphs
We study the computational complexity of a specific type of graph planarization. For a fixed ℓ, we say that a graph G is ℓ-subgraph contractible if there exist vertex disjoint subgraphs of G of size at most ℓ each such that after contracting these subgraphs into single vertices the graph becomes planar. When ℓ = 2, this is equivalent to asking if there exists a matching in G whose contraction makes the graph planar. We say in this case that G is matching contractible. We show that matching contractibility is NP-complete and give a quadratic algorithm for the version parameterized by the number k of edges to be contracted. Our results generalize for ℓ ≥ 2 and we also show that ℓ-subgraph contractibility remains NP-complete when we contract to graphs of higher genus g (instead of planar graphs).
A linear-time algorithm for four-partitioning four-connected planar graphs
Information Processing Letters, 1997
Given a graph G = (YE), four distinct vertices uI.u2,u3&4 E V and four natural numbers nl , n2, n3. ?Q such that cf=, ni = IV\, we wish to find a partition VI, &, 6, & of the vertex set V such that ui E x, 1x1 = ni and L$ induces a connected subgraph of G for each i. 1 < i < 4. In this paper we give a simple linear-time algorithm to find such a partition if G is a 4-connected planar graph and ~1. ~2. 143 and u4 are located on the same face of a plane embedding of G. Our algorithm is based on a "4canonical decomposition" of G, which is a generalization of an St-numbering and a "canonical 4-ordering" known in the area of graph drawings. @ 1997 Elsevier Science B.V.
Complexity and approximability of the k -way vertex cut
Networks, 2013
In this article, we consider k-way vertex cut: the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k-way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed-parameter tractable on planar graphs with the size of the separator as the parameter.
A linear time algorithm for graph partition problems
Information Processing Letters, 1992
Parameterized complexity: exponential speed-up for planar graph problems
Journal of Algorithms, 2004
We discuss general techniques, centered around the "Layerwise Separation Property" (LSP) of a planar graph problem, that allow to develop algorithms with running time c √ k |G|, given an instance G of a problem on planar graphs with parameter k. Problems having LSP include planar vertex cover, planar independent set, and planar dominating set. Extensions of our speed-up technique to basically all fixed-parameter tractable planar graph problems are also exhibited. Moreover, we relate, e.g., the domination number or the vertex cover number, with the treewidth of a plane graph. √ k n O(1) for constant c. Moreover, we discuss an extension of our technique which applies to basically all fixed-parameter tractable graph problems.
A note on vertex partitions
We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lovasz, Catlin, Kostochka and Rabern.
NP-completeness of the Planar Separator Problems
Journal of Graph Algorithms and Applications, 2006
For a given graph G, the Separator Problem asks whether a vertex or edge set of small cardinality (or weight) exists whose removal partitions G into two disjoint graphs of approximately equal sizes. Called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set, both problems are NP-complete for general unweighted graphs [6]. Despite the significance of planar graphs, it has not been known whether the Planar Separator Problem, which considers a planar graph and a threshold as an input, is NP-complete or not. In this paper, we prove that the Vertex Separator Problem is in fact NP-complete when G is a vertex weighted planar graph. The Edge Separator Problem will be shown NP-complete when G is a vertex and edge weighted planar graph. In addition, we consider how to treat the constant α ∈ R + of the α-Separator Problem that partitions G into two disjoint graphs of size at most (1 − α) |V (G)|. The α-Separator Problem is not NP-complete for all real numbers α ∈ (0, 1/2], because it would imply uncountably many Non Deterministic Turing Machines. We will present a general scheme for treating a constant in computer arithmetic, by introducing the notion of real numbers comparable with rationals in polynomial time. This approach allows us to prove NP-completeness for each such real number α.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.