On the (Parameterized) Complexity of Recognizing Well-covered ( r , ℓ )-graph

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ)wc-g for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered(r,ℓ)(r,ℓ)-Graph problem (wc-(r,ℓ)(r,ℓ)g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition, and the question is whether G is well-covered. This generates two infinite families of problems, for any fixed non-negative integers r and ℓ, which we classify as being P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc-(r,0)(r,0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size α of a maximum independent set of the input graph, its neighborhood diversity, its clique-width, or the number ℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of determining whether every maximal independent set of an input graph G has cardinality equal to k can be reduced to the wc-(0,ℓ)(0,ℓ)g problem parameterized by ℓ. In addition, we prove that both problems are coW[2]-hard but can be solved in XP-time

On the (Parameterized) Complexity of Recognizing Well-covered (r,l)-graphs

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ) wcg for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered (r,ℓ)(r,ℓ)-Graph problem (wc (r,ℓ)(r,ℓ) g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition of V(G) into r independent sets and ℓℓ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ℓℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,ℓ)(0,ℓ) g problem parameterized by ℓℓ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP⊆NP/polycoNP⊆NP/poly

A complexity approach of the (r,l)-well-covered graphs: recognition, sandwich problems and probe

[EN] A (r,l)-partition of a graph G is a partition of its vertex set into r independent sets and cliques. A graph is a (r,l)-graph if it admits a (r,l)-partition. A graph is a (r,l) -graph if it admits a (r,l)-partition. A graph is well-covered when each maximal independent set is maximum. A graph is a (r, l)-well-covered graph if it is (r,l) and well-covered, simultaneously. In this work we consider two different decision problems. In the (r,l)-well-covered graph problem (gbc(r,l) for short), a graph G is provided as input, and the question is whether G is an (r,l)- well-covered graph. In the well-covered (r,l)-graph problem (g(r,l )bc for short), a (r,l)-graph G together with an (r,l)-partition of V (G) into r independent sets and cliques are provided as input, and the question is whether G is wellcovered. In the context of sandwich problems, we consider the classes (r,l)-well-covered which are recognized in polynomial time, namely: (0, 1), (1, 0), (0, 2), (2, 0), (1, 1), and (1, 2). We solved this problem for five of those six classes, and the problem remains open only when (r,l) = (2, 0). We also present, in this work, the solution of partitioned probe for (r,l)-wellcovered graphs problem for all graph classes well covered-(r,l) which are recognizable in polynomial time, except for the classes (2, 0) and (1, 2). In addition, we consider the parameterized complexity of well-covered graph problem with special emphasis on the case where the given graph is a (r,l)-graph for several choices of parameters, such as the size α of a maximal independent set of the input graph, neighborhood diversity, and the number of cliques in an (r,l)-partition.[PT] Uma partição-(r,l) de um grafo G é uma partição do seu conjunto de vértices em r conjuntos independentes e ` cliques. Um grafo é chamado de grafo-(r,l) se admite uma partição-(r,l). Um grafo é bem coberto se todo conjunto independente maximal é máximo. Um grafo é um grafo bem coberto-(r,l) se é, ao mesmo tempo, (r,l) e bem coberto. Neste trabalho consideramos dois tipos de problemas de decisão distintos. No problema grafos bem cobertos-(r,l) (abreviadamente gbc(r,l)), o grafo G é dado, e quer-se decidir se o grafo G é um grafo-(r,l) bem coberto. No problema grafos-(r,l) bem cobertos (abreviadamente g(r,l) bc) é dado um grafo-(r,l) como entrada juntamente com uma partição de V (G) em r conjuntos independentes e cliques, e a pergunta é se G é bem coberto. No contexto dos problemas sanduíche, consideramos a classe dos grafos bem cobertos-(r,l) que são reconhecidos em tempo polinomial, a saber: (0, 1), (1, 0), (0, 2), (2, 0), (1, 1) e (1, 2). Resolvemos este problema para cinco das seis classes, e o problema permanece em aberto apenas quando (r,l) = (2, 0). Também apresentamos, neste trabalho, a solução do problema probe particionado para grafos bem cobertos-(r,l) para todas as classes de grafos bem cobertos- (r,l) que são reconhecíveis em tempo polinomial, com exceção das classes (2, 0) e (1, 2). Além disso, consideramos a complexidade parametrizada do problema grafo bem coberto, com ênfase especial no caso em que o grafo dado é um grafo-(r,l), para algumas escolhas de parâmetros, tais como o tamanho α de um conjunto independente maximal do grafo de entrada, diversidade de vizinhança, e o número de cliques em uma partição-(r,l).