On the recognition of fuzzy circular interval graphs

The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph, and even admits a polynomial kernel when H is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs.Complementing the above positive results, we prove W[1]-hardness of Induced Graph Matching on graphs excluding K 1,4 as an induced subgraph, for any fixed complete graph H. In particular, we show that Independent Set is W[1]-hard on K 1,4 -free graphs.Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or NP-complete, depending on the connectivity of H and the structure of G.

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“…We recall a result by Oriolo et al [41], who showed that a thickening of a circular interval trigraph can be recognized in polynomial time.…”

Section: Special Trigraphsmentioning

confidence: 78%

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

Hermelin

Mnich²,

Leeuwen

2014

Algorithmica

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“…Given two adjacent vertices that are non-universal to each other, a simple algorithm recognizes in O(n 2 )time whether they have a PH-embedding. This routine, which we report below, was independently proposed by King and Reed [10] and Pietropaoli [14] (see also [13]). Actually King and Reed designed an algorithm for a slightly different problem: call {K 1 , K 2 } a non-trivial homogeneous (NTH) pair of cliques in G if {K 1 , K 2 } is a homogeneous pair of cliques in G, and G[K 1 ∪ K 2 ] has an induced C 4 .…”

Section: Preliminariesmentioning

confidence: 98%

“…We now move from graph invariants to graph properties. First, Oriolo, Pietropaoli, and Stauffer [13] show that a suitable reduction of PH pairs of cliques preserves the property of a graph of being, or not being, a fuzzy circular interval graph, and they exploit this fact in an algorithm for recognizing fuzzy circular interval graphs. Their reduction can also be embedded in our framework.…”

Section: Preserving Some Graph Invariant or Propertymentioning

confidence: 99%

“…We say that a pair of cliques {K 1 , K 2 } is proper if each vertex in K 1 is neither complete nor anticomplete to K 2 , and each vertex in K 2 is neither complete nor anticomplete to K 1 . Those reduction techniques are designed to preserve graph invariants, such as chromatic number [8,10] and stability number [12], or graph properties, such as the property of a graph of being quasi-line [2], fuzzy circular interval [13], or even facets of the stable set polytope [7]. The state of the art complexity for recognizing whether a graph G(V, E) has some proper and homogeneous pairs of cliques is O(|V (G)| 2 |E(G)|) [10,14].…”

Section: Introductionmentioning

confidence: 99%

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A fast algorithm to remove proper and homogeneous pairs of cliques (while preserving some graph invariants)

Faenza

Oriolo

Snels

2011

Operations Research Letters

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We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at most |E(G)| steps builds a new graph G ′ without proper and homogeneous pairs of cliques, and such that G and G ′ agree on the value of some relevant invariant (or property).

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“…Now find a representation of G, using the algorithm of Oriolo, Pietropaolo, and Stauffer [25]. This gives a representation A of G as described in Lemma A.2.…”

Section: Proofmentioning

confidence: 99%

Parameterized Complexity of Induced H-Matching on Claw-Free Graphs

Hermelin

Mnich

Leeuwen

2012

Algorithms – ESA 2012

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The Induced H-Matching problem asks to find k disjoint, induced subgraphs isomorphic to H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced H-Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph of constant size, and even admits a polynomial kernel when H is a clique. Both results rely on a new, strong algorithmic structure theorem for claw-free graphs. To show the fixed-parameter tractability of the problem, we additionally apply the color-coding technique in a novel way. Complementing the above two positive results, we prove the W[1]-hardness of Induced H-Matching for graphs excluding K 1,4 as an induced subgraph. In particular, we show that Independent Set is W[1]-hard on K 1,4 -free graphs.

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On the recognition of fuzzy circular interval graphs

Cited by 8 publications

References 17 publications

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

A fast algorithm to remove proper and homogeneous pairs of cliques (while preserving some graph invariants)

Parameterized Complexity of Induced H-Matching on Claw-Free Graphs

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