A characterization of chain probe graphs

Golumbic, Martin Charles; Maffray, Frédéric; Morel, Grégory

doi:10.1007/s10479-009-0584-6

Cited by 10 publications

(7 citation statements)

References 15 publications

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“…In fact, as pointed out in (Golumbic et al 2009), an O(n 2 )-algorithm can be derived from the proof of Theorem 1.…”

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confidence: 89%

“…The second one characterizes these graphs via a certain "enhanced graph": G is a chain partitioned probe graph if and only if the enhanced graph G * is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m 2 )-time recognition algorithm for chain partitioned probe graphs.Given a graph class C, a graph G = (V , E) is called a C probe graph if there exists an independent set S ⊆ V and a set E ⊆ S 2 such that the graph G = (V , E ∪ E ) is in the class C, where S 2 stands for the set of all 2-element subsets of S. Probe graphs have been investigated for various graph classes; see Golumbic et al (2009) and the literature given there for more information.In case of bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges), also called chain graphs by Yannakakis (1981) or difference graphs by Hammer et al (1990), Golumbic et al (2009) very recently found a characterization of chain probe graphs as follows; notice that chain probe graphs are necessarily bipartite. Theorem 1 (Golumbic et al 2009) Let G be a bipartite graph.…”

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confidence: 99%

“…Given a graph class C, a graph G = (V , E) is called a C probe graph if there exists an independent set S ⊆ V and a set E ⊆ S 2 such that the graph G = (V , E ∪ E ) is in the class C, where S 2 stands for the set of all 2-element subsets of S. Probe graphs have been investigated for various graph classes; see Golumbic et al (2009) and the literature given there for more information.…”

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confidence: 99%

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Two characterizations of chain partitioned probe graphs

Lê

2010

Ann Oper Res

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Chain graphs are exactly bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges). A graph G = (V , E) with a given independent set S ⊆ V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain "enhanced graph": G is a chain partitioned probe graph if and only if the enhanced graph G * is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m 2 )-time recognition algorithm for chain partitioned probe graphs.Given a graph class C, a graph G = (V , E) is called a C probe graph if there exists an independent set S ⊆ V and a set E ⊆ S 2 such that the graph G = (V , E ∪ E ) is in the class C, where S 2 stands for the set of all 2-element subsets of S. Probe graphs have been investigated for various graph classes; see Golumbic et al. (2009) and the literature given there for more information.In case of bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges), also called chain graphs by Yannakakis (1981) or difference graphs by Hammer et al. (1990), Golumbic et al. (2009) very recently found a characterization of chain probe graphs as follows; notice that chain probe graphs are necessarily bipartite. Theorem 1 (Golumbic et al. 2009) Let G be a bipartite graph. Then G is a chain probe graph if and only if it is {C 6 , P 7 , 3K 2 , H 1 , H 2 , H 3 }-free.

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“…In fact, as pointed out in (Golumbic et al 2009), an O(n 2 )-algorithm can be derived from the proof of Theorem 1.…”

mentioning

confidence: 89%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Two characterizations of chain partitioned probe graphs

Lê

2010

Ann Oper Res

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“…A characterization of chain probe graphs and an O(n 2 ) recognition algorithm was given in Golumbic et al (2009). The partitioned chain probe graph problem assumes that the independent set S is given and fixed in advance, and is a special case of the chain sandwich problem where E 2 \ E 1 = S × S. The complexity of the partitioned chain probe problem has also been shown to be polynomial in Van Bang (2010).…”

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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“…Chordal probe graphs were investigated in [11] and a characterization and recognition algorithm was given in [2] for both the partitioned and non-partitioned versions. Recently, characterizations and recognitions have been given for chain probe graphs in [12], and for threshold probe graphs and trivially-perfect probe graphs, also known as quasi-threshold probe graphs, in [4]. Other probe classes are to be found in [5,6].…”

Section: Introductionmentioning

confidence: 99%