Algorithms for unipolar and generalized split graphs

Eschen, Elaine M.; Wang, Xiaoqiang

doi:10.1016/j.dam.2013.08.011

Cited by 19 publications

(25 citation statements)

References 39 publications

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“…For colourings, we explicitly find a minimum colouring in each co-bipartite graph G[C 0 ∪ C i ], and such colourings can be fitted together using no more colours, since C 0 is a clique cutset. Assume that G[C 0 ∪ C i ] contains n i vertices and m i edges, so each n i ≤ n and The approach of Eschen and Wang [5] is very similar, and they give more details, but unfortunately there is a problem with their analysis, and a corrected version of their analysis yields O(n 3.5 / log n) time, instead of the claimed O(n 2.5 / log n). In order to see the problem consider the case when the input graph is a split graph with an equitable partition.…”

Section: Algorithms For Random Perfect Graphsmentioning

confidence: 99%

“…Perfect graphs can be recognised in polynomial time [2,3], and there are many NP-hard problems which are solvable in polynomial time for perfect graphs, including the stable set problem, the clique problem, the colouring problem, the clique covering problem and their weighted versions [4]. If the input graph is restricted to be a generalised split graph, then there are much more efficient algorithms for the problems above [5]. In this paper we address the problem of efficiently recognising generalised split graphs, and finding a witnessing partition.…”

Section: Definition and Motivationmentioning

confidence: 99%

“…Eschen and Wang [5] show that, given a generalised split graph G with n vertices together with a unipolar representation of G or G, we can efficiently solve each of the following four problems: find a maximum clique, find a maximum independent set, find a minimum colouring, and find a minimum clique cover.…”

Section: Algorithms For Random Perfect Graphsmentioning

confidence: 99%

“…Previous recognition algorithms for unipolar graphs include [6] which achieves O(n 3 ) running time, [7] with O(n 2 m) time and [5] with O(nm + nm ) time, where n and m are respectively the number of vertices and edges of the input graph, and m is the number of edges added after a triangulation of the input graph. Note that almost all unipolar graphs and almost all generalised split graphs have (1 + o(1))n 2 /4 edges (see our paper "Random perfect graphs", which is under preparation).…”

Section: Definition and Motivationmentioning

confidence: 99%

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Recognition of Unipolar and Generalised Split Graphs

McDiarmid

Yolov

2015

Algorithms

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A graph is unipolar if it can be partitioned into a clique and a disjoint union of cliques, and a graph is a generalised split graph if it or its complement is unipolar. A unipolar partition of a graph can be used to find efficiently the clique number, the stability number, the chromatic number, and to solve other problems that are hard for general graphs. We present an O(n 2 )-time algorithm for recognition of n-vertex generalised split graphs, improving on previous O(n 3 )-time algorithms.

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Section: Algorithms For Random Perfect Graphsmentioning

confidence: 99%

Section: Definition and Motivationmentioning

confidence: 99%

Section: Algorithms For Random Perfect Graphsmentioning

confidence: 99%

Section: Definition and Motivationmentioning

confidence: 99%

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Recognition of Unipolar and Generalised Split Graphs

McDiarmid

Yolov

2015

Algorithms

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“…Determining if a graph G contains a perfect code is NP-complete, even for many restricted classes of graphs, such as planar graphs with maxi¬mum degree three [43,24] and many others [23]. Efficient algorithms exists to determine if some classes of graphs contain a perfect code: trees [24], circular-arc graphs [41], and others [14,45].…”

Section: Perfect Codesmentioning

confidence: 99%

A Taxonomy of Perfect Domination

Klostermeyer¹

2015

Journal of Discrete Mathematical Sciences and Cryptography

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A decomposition approach to solve the selective graph coloring problem in some perfect graph families

2018

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Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP-hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state-of-the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low-density graphs, and regardless of edge-density in the class of chordal graphs. the number of colors necessary to color the vertices in the selection is minimum. In a graph where each cluster consists of a single vertex, SEL-COL becomes equivalent to the classical graph coloring problem [5]. SEL-COL, or partition coloring as it is alternatively called in the literature, is motivated by the wavelength routing and assignment problem, and was introduced in Li and Simha [17]. There, a telecommunication network, which is a collection of terminal nodes linked by optical fibers capable of carrying a certain number of wavelengths, and a set of source-destination node pairs are given. The problem is concerned with finding the minimum number of distinct wavelengths to assign to each route, that is, paths connecting the given source-destination pairs, such that no two routes that have a common link are assigned the same wavelength. In this setting, the set of all possible routes and pairs of routes possessing a common link correspond to vertices and edges respectively, and groups of routes connecting a given source-destination pair constitute the clusters in the host graph. Then, selection of one vertex per cluster achieves the goal of finding a route between each pair of terminals, and coloring of the selection delivers a proper wavelength to each route.In the standard graph coloring problem, assignments (colors) on the entities (vertices) are done without any regard to alternative choices for them. However, as the example applications reveal, there are cases where entities have their own set of feasible options (clusters) and thus should be allocated one among those alternatives. In such cases, where the basic graph coloring framework fails to suffice, SEL-COL bridges the gap by offering the required flexibility.SEL-COL ha...

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Algorithms for unipolar and generalized split graphs

Cited by 19 publications

References 39 publications

Recognition of Unipolar and Generalised Split Graphs

Recognition of Unipolar and Generalised Split Graphs

A Taxonomy of Perfect Domination

A decomposition approach to solve the selective graph coloring problem in some perfect graph families

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