A linear-time algorithm for k-partitioning doughnut graphs

HAL (Le Centre pour la Communication Scientifique Directe), 1994

Theoretical Computer Science, 1998

This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements (vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases (called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called R-PART-WOB). In this paper, we show that K-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1) for any k 22, R-PART-WOB can be solved in O(l~I~~fI~I) t' lme for every 4-edge-connected graph G = (V,E), (2) 3-PART-WOB can be solved in 0(1 VI') for every 2-edge-connected graph G = (V,E) and (3) 4-PART-WOB can be solved in O(lE1*) for every 3-edge-connected graph G =(V,E).

The American Mathematical Monthly, 2015

Given a graph G on n vertices, for which m is it possible to partition the edge set of the m-fold complete graph mK n into copies of G? We show that there is an integer m 0 , which we call the partition modulus of G, such that the set M (G) of values of m for which such a partition exists consists of all but finitely many multiples of m 0. Trivial divisibility conditions derived from G give an integer m 1 which divides m 0 ; we call the quotient m 0 /m 1 the partition index of G. It seems that most graphs G have partition index equal to 1, but we give two infinite families of graphs for which this is not true. We also compute M (G) for various graphs, and outline some connections between our problem and the existence of designs of various types.

2011

Graph partitioning is a classical graph theory problem that has proven to be NP-hard.

Information Processing Letters, 1992

Lecture Notes in Computer Science, 1999

Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , A k with ||Ai| − |Aj|| ≤ 1. We consider the problem of determining the minimal size of such a cut for a given graph. For this we introduce a new lower bound method which is based on the solution of an extremal set problem and present bounds for some graph classes based on Hamming graphs.

Journal of Graph Theory, 2015

Given k ≥ 1, a k-proper partition of a graph G is a partition P of V (G) such that each part P of P induces a k-connected subgraph of G. We prove that if G is a graph of order n such that δ(G) ≥ √ n, then G has a 2-proper partition with at most n/δ(G) parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If G is a graph of order n with minimum degree δ(G) ≥ c(k − 1)n, where c = 2123 180 , then G has a k-proper partition into at most cn δ(G) parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint k-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760-764] and both the degree condition and the number of parts are best possible up to the constant c.

2021

Partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph G = (V,E) and a weight function w : V → Q≥, a connected k-partition of G is a partition of V such that each class induces a connected subgraph. The balanced connected k-partition problem consists in finding a connected k-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected k-partitions that either maximize the weight of a lightest class (max-min BCPk) or minimize the weight of a heaviest class (min-max BCPk). Such problems are equivalent when k = 2, but they are different when k ≥ 3. In this work, we propose a simple pseudo-polynomial k 2 -approximation algorithm for min-max BCPk which runs in time O(W |V ||E|), where W = ∑ v∈V w(v). Based on this algorithm and using a scaling technique, we design a (polynomial) ...

Combinatorica, 2007

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/ √ lg n). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C · cp(G)/ √ lg n classes, then NP ⊆ RTime(n O(lg lg n) ). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lg n) c ) for some constant c strictly between 0 and 1. * The work reported here is a merger of the results reported in [KRS05] and .

2002