Helly property, clique graphs, complementary graph classes, and sandwich problems

Dourado, Mitre C.; Petito, Priscila; Teixeira, Ricardo Rodrigues; Figueiredo, Celina M. H. de

doi:10.1590/s0104-65002008000200004

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…The recognition problem for a class of graphs C is equivalent to the particular graph sandwich problem where E 1 = E 2 , that is, the optional edge set is empty. Graph sandwich problems have attracted much attention lately because of many applications and by the fact that they naturally generalize recognition problems [2][3][4][5][6]. Note that −SP is clearly at least as hard as the problem of recognizing graphs with property , since given a polynomial-time algorithm for −SP, it is possible to use this algorithm with E 1 = E 2 = E to recognize if a graph G = (V , E) satisfies property .…”

Section: Graph Sandwich Problem For Propertymentioning

confidence: 99%

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

2014

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Background: In this work, we consider the graph sandwich decision problem for property , introduced by Golumbic, Kaplan and Shamir: given two graphs G 1 = (V, E 1) and G 2 = (V, E 2), the question is to know whether there exists a graph G = (V, E) such that E 1 ⊆ E ⊆ E 2 and G satisfies property. Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties : 'G is a chordal-(k,)-graph' and 'G is a strongly chordal-(k,)-graph', for all k,. Methods: In order to do that, we consider each pair of positive values of k and , exhibiting correspondent polynomial algorithms, or NP-complete reductions. Results: We prove that the STRONGLY CHORDAL-(k,) GRAPH SANDWICH PROBLEM is NP-complete, for k ≥ 1 and ≥ 1, and that the CHORDAL-(k,) GRAPH SANDWICH PROBLEM is NP-complete, for positive integers k and such that k + ≥ 3. Moreover, we prove that both problems are in P when k or is zero and k + ≤ 2. Conclusions: To complete the complexity dichotomy concerning these problems for all nonnegative values of k and , there still remains the open question of settling the complexity for the case k + ≥ 3 and one of them is equal to zero.

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Section: Graph Sandwich Problem For Propertymentioning

confidence: 99%

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

2014

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Section: Graph Sandwich Problem For Propertymentioning

confidence: 99%

The chain graph sandwich problem

et al. 2010

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The chain graph sandwich problem asks: Given a vertex set V , a mandatory edge set E 1 , and a larger edge set E 2 , is there a graph G = (V , E) such that E 1 ⊆ E ⊆ E 2 with G being a chain graph (i.e., a 2K 2 -free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E 1 is a connected graph, which has a linear-time solution, (2) the threshold graph sandwich problem, which has a linear-time solution, and (3) the chain probe graph problem, which has a polynomial-time solution.

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“…Hence, any sandwich graph G = (V, E) for the pair G 1 , G 2 must contain all mandatory edges and no forbidden edges. Graph sandwich problems have drawn much attention because they naturally generalize graph recognition problems and have many applications [8,9,10,14,19].…”

Section: Introductionmentioning

confidence: 99%

(k, ℓ)-Sandwich Problems: why not ask for special kinds of bread?

Couto,

Faria,

Klein

et al. 2012

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In this work, we consider the Golumbic, Kaplan, and Shamir decision sandwich problem for a property Π: given two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ), the question is: Is there a graph G = (V, E) such that E 1 ⊆ E ⊆ E 2 and G satisfies Π? The graph G is called sandwich graph. Note that what matters here is just the "filling" of the sandwich. Our proposal is to try different kinds of "bread" for each chosen special sandwich filling. In other words, we focus on the complexity of sandwich problems when, beforehand, it is known that G i satisfies a property Π i , i = 1, 2. Let (Π 1 , Π, Π 2 )-sp denote the sandwich problem for property Π when G i satisfies Π i , called sandwich problem with boundary conditions. When G i is not required to satisfy any special property, Π i is denoted by * . A graph G is (k, ℓ) if there is a partition of V (G) into k independent sets and ℓ cliques. It is known that ( * , (k, ℓ), * )-sp is NP-complete, for all k + ℓ greater than 2. In order to motivate this new work proposal, in this paper we describe polynomial-time algorithms for three sandwich problems with boundary conditions: (perfect, (k, ℓ), poly-

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Helly property, clique graphs, complementary graph classes, and sandwich problems

Cited by 3 publications

References 12 publications

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

The chain graph sandwich problem

(k, ℓ)-Sandwich Problems: why not ask for special kinds of bread?

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