Selective Graph Coloring in Some Special Classes of Graphs

Demange, Marc; Monnot, Jérôme; Pop, Petrică C.; Ries, Bernard

doi:10.1007/978-3-642-32147-4_29

Cited by 6 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another perspective is to study the complexity and approximation of the Empire Problem for other classes of sparse planar graphs with small average degree like nK 3 and nK 4 .…”

Section: Resultsmentioning

confidence: 99%

“…It consists of selecting one vertex per empire (not all the empire) in such a way that the chromatic number of the resulting induced selection is the smallest possible. In [4], some complexity results for various classes of graphs are given. Hence, the decision version of the selective graph coloring problem can be viewed as the restriction of the Empire Problem to select one vertex per empire.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complexity Results for the Empire Problem in Collection of Stars

Couëtoux

Monnot

Toubaline

2012

Combinatorial Optimization and Applications

View full text Add to dashboard Cite

Abstract. In this paper, we study the Empire Problem, a generalization of the coloring problem to maps on two-dimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of deciding if s colors are sufficient to color the vertices of the graph such that vertices of the same block have the same color and vertices of two adjacent blocks have different colors. In this paper, we prove that given a 5-regular graph, deciding if there exists a 4-coloration is NP-complete. Also, we propose conditional NP-completeness results for the Empire Problem when the graph is a collection of stars. A star is a graph isomorphic to K1,q for some q ≥ 1. More exactly, we prove that for r ≥ 2, if the (2r − 1)-coloring problem in 2r-regular connected graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = 2r − 1 is NP-complete for forests of K1,r. Moreover, we prove that this result holds for r = 2. Also for r ≥ 3, if the r-coloring problem in (r + 1)-regular graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = r is NP-complete for forests of K1,1 = K2, i.e., forest of edges. Additionally, we prove that this result is valid for r = 2 and r = 3. Finally, we prove that these results are the best possible, that is for smallest value of s or r, the Empire Problem in these classes of graphs becomes polynomial.

show abstract

“…Another perspective is to study the complexity and approximation of the Empire Problem for other classes of sparse planar graphs with small average degree like nK 3 and nK 4 .…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complexity Results for the Empire Problem in Collection of Stars

Couëtoux

Monnot

Toubaline

2012

Combinatorial Optimization and Applications

View full text Add to dashboard Cite

show abstract

“…Many other interesting applications of the selective graph coloring problem related for instance to timetabling and dichotomy-based constraint encoding can be found in [5]. Recently, the selective coloring problem was shown to be NP-hard in paths, cycles and split graphs even with strong restrictions on the cardinality of the clusters [6].…”

Section: Introductionmentioning

confidence: 99%

Perfectness of clustered graphs

Bonomo

Cornaz

Ekim

et al. 2013

Discrete Optimization

Self Cite

View full text Add to dashboard Cite

“…Demange et al [10,11] considered this type of graph coloring problem in the framework of generalized network design problems and named it selective graph coloring problem. They investigated as well some special classes of graphs including split graphs, bipartite graphs and q-partite graphs and settled the complexity status of the PGCP in these particular classes.…”

mentioning

confidence: 99%

An improved hybrid ant-local search algorithm for the partition graph coloring problem

Fidanova

Pop

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tIn this paper we propose a hybrid Ant Colony Optimization (ACO) algorithm for the Partition Graph Coloring Problem (PGCP). Given an undirected graph G = (V , E), whose nodes are partitioned into a given number of the sets, the goal of the PGCP is to find a subset V * ⊂ V that contains exactly one node from each cluster and a coloring for V * so that in the graph induced by V * , two adjacent nodes have different colors and the total number of used colors is minimal. Our hybrid algorithm is obtained by executing a local search procedure after every ACO iteration. The performance of our algorithm is evaluated on a set of instances commonly used as benchmark and the computational results show that compared to state-of-the-art algorithms, our improved hybrid ACO algorithm achieves solid results in very short run-times.

show abstract

Selective Graph Coloring in Some Special Classes of Graphs

Cited by 6 publications

References 17 publications

Complexity Results for the Empire Problem in Collection of Stars

Complexity Results for the Empire Problem in Collection of Stars

Perfectness of clustered graphs

An improved hybrid ant-local search algorithm for the partition graph coloring problem

Contact Info

Product

Resources

About