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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
A LinearTime Algorithm for FourPartitioning FourConnected Planar Graphs (Extended Abstract)
 143
, 1997
"... Given a graph G = (V; E), k distinct vertices u 1 ; u 2 ; 1 1 1, u k 2 V and k natural numbers n 1 ; n 2 ; 1 1 1 ; n k such that P k i=1 n i = jV j, we wish to find a partition V 1 ; V 2 ; 1 1 1 ; V k of the vertex set V such that u i 2 V i , jV i j = n i , and V i induces a connected subgraph of G ..."
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Given a graph G = (V; E), k distinct vertices u 1 ; u 2 ; 1 1 1, u k 2 V and k natural numbers n 1 ; n 2 ; 1 1 1 ; n k such that P k i=1 n i = jV j, we wish to find a partition V 1 ; V 2 ; 1 1 1 ; V k of the vertex set V such that u i 2 V i , jV i j = n i , and V i induces a connected subgraph of G for each i, 1 i k. Such a partition is called a k partition of G. The problem of finding a kpartition of a general graph is NPhard [DF85], and hence it is very unlikely that there is a polynomialtime algorithm to solve the problem. Although not every graph has a kpartition, Gyori and Lov'asz independently proved that every kconnected graph has a kpartition for any u 1 ; u 2 ; 1 1 1 ; u k and n 1 ; n 2 ; 1 1 1 ; n k [G78, L77]. However, their proofs do not yield any polynomialtime algorithm for actually finding a k ...
Computational Balloon Twisting: The Theory of Balloon Polyhedra
, 2008
"... This paper builds a general mathematical and algorithmic theory for balloontwisting structures, from balloon animals to balloon polyhedra, by modeling their underlying graphs (edge skeleta). In particular, we give algorithms to find the fewest balloons that can make exactly a desired graph or, usin ..."
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This paper builds a general mathematical and algorithmic theory for balloontwisting structures, from balloon animals to balloon polyhedra, by modeling their underlying graphs (edge skeleta). In particular, we give algorithms to find the fewest balloons that can make exactly a desired graph or, using fewer balloons but allowing repeated traversal or shortcuts, the minimum total length needed by a given number of balloons. In contrast, we show NPcompleteness of determining whether such an optimal construction is possible with balloons of equal length.
Efficient Algorithms for Drawing Planar Graphs
, 1999
"... x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ................. ..."
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x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Boxrectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...
Fully decomposable split graphs
"... Abstract. We discuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable. ..."
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Abstract. We discuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable.
Manuscript Bisecting a FourConnected Graph with Three Resource Sets
"... Let G =(V,E) be an undirected graph with a node set V andanarcsetE. G has k pairwise disjoint subsets T1,T2,...,Tk of nodes, called resource sets, where Ti  is even for each i. The partition problem with k resource sets asks to find a partition V1 and V2 of the node set V such that the graphs indu ..."
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Let G =(V,E) be an undirected graph with a node set V andanarcsetE. G has k pairwise disjoint subsets T1,T2,...,Tk of nodes, called resource sets, where Ti  is even for each i. The partition problem with k resource sets asks to find a partition V1 and V2 of the node set V such that the graphs induced by V1 and V2 are both connected and V1 ∩ Ti  = V2 ∩ Ti  = Ti/2holds for each i =1, 2,...,k. The problem of testing whether such a bisection exists is known to be NPhard even in the case of k = 1. On the other hand, it is known that if G is (k +1)connected for k =1, 2, then a bisection exists for any given resource sets, and it has been conjectured that for k ≥ 3, a (k + 1)connected graph admits a bisection. In this paper, we show that for k = 3, the conjecture does not hold, while if G is 4connected and has K4 as its subgraph, then a bisection exists and it can be found in O(V  3 log V ) time. Moreover, we show that for an arcversion of the problem, the (k + 1)edgeconnectivity suffices for k =1, 2, 3. Key words: graph algorithm, graph partition problem, graph connectivity, embedding ∗ Corresponding author.
On three polynomial kernels of sequences for arbitrarily partitionable
, 2013
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"... Noname manuscript No. (will be inserted by the editor) On the complexity of partitioning a graph into a few connected subgraphs ..."
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Noname manuscript No. (will be inserted by the editor) On the complexity of partitioning a graph into a few connected subgraphs
A LinearTime Algorithm for kPartitioning Doughnut Graphs MD. REZAUL KARIM 1 KAISER MD. NAHIDUZZAMAN 2
, 2009
"... Abstract. Given a graph G = (V, E), k natural numbers n1, n2,..., nk such that ∑ k i=1 ni = V , we wish to find a partition V1, V2,..., Vk of the vertex set V such that Vi  = ni and Vi induces a connected subgraph of G for each i, 1 ≤ i ≤ k. Such a partition is called a kpartition of G. The pr ..."
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Abstract. Given a graph G = (V, E), k natural numbers n1, n2,..., nk such that ∑ k i=1 ni = V , we wish to find a partition V1, V2,..., Vk of the vertex set V such that Vi  = ni and Vi induces a connected subgraph of G for each i, 1 ≤ i ≤ k. Such a partition is called a kpartition of G. The problem of finding a kpartition of a graph G is NPhard in general. It is known that every kconnected graph has a kpartition. But there is no polynomial time algorithm for finding a kpartition of a kconnected graph. In this paper we give a simple lineartime algorithm for finding a kpartition of a “doughnut graph ” G.
Approximation and Inaproximability Results on Balanced Connected Partitions of Graphs
"... Let G = (V, E) be a connected graph with a weight function w: V → Z+ and let q ≥ 2 be a positive integer. For X ⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a qpartition P = (V1, V2,..., Vq) of V such that G[Vi] is connected (1 ≤ i ≤ ..."
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Let G = (V, E) be a connected graph with a weight function w: V → Z+ and let q ≥ 2 be a positive integer. For X ⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a qpartition P = (V1, V2,..., Vq) of V such that G[Vi] is connected (1 ≤ i ≤ q) and P maximizes min{w(Vi) : 1 ≤ i ≤ q}. This problem is called Max Balanced Connected qPartition and is denoted by BCPq. We show that for q ≥ 2 the problem BCPq is NPhard in the strong sense, even on qconnected graphs, and therefore does not admit a FPTAS, unless P = NP. We also show another inapproximability result for BCP2. For the problemapproximation algorithm obtained by Chlebíková; for q = 3 and q = 4 we present 2approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called Max Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless P = NP. BCPq restricted to qconnected graphs, it is known that for q = 2 the best result is a 4 3