## Permanents, Pfaffian Orientations, and Even Directed Circuits (1999)

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Citations: | 57 - 13 self |

### BibTeX

@MISC{Robertson99permanents,pfaffian,

author = {Neil Robertson and P. D. Seymour and Robin Thomas},

title = {Permanents, Pfaffian Orientations, and Even Directed Circuits },

year = {1999}

}

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### Abstract

Given a 0-1 square matrix A, when can some of the 1’s be changed to −1’s in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign, or are both zero) is non-singular? When is a hypergraph with n vertices and n hyperedges minimally non-bipartite? When does a bipartite graph have a “Pfaffian orientation”? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all the above problems are equivalent. We prove a structural char-acterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic non-planar bipartite graph.

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Citation Context ...directing every edge from A to B, and contracting every edge in M. The problem is equivalent to finding the strongly connected components of D, which is well-known, and is described, for instance, in =-=[2]-=-. 9.2. Algorithm. Input. A connected brace G on n vertices, and a list L of all trisectors of G. Output. Either a Pfaffian orientation of G, or a valid statement that G has no Pfaffian orientation. Ru... |

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Citation Context ...tation of G, or a valid statement that G has no Pfaffian orientation. Running time. O(n 3 ).974 NEIL ROBERTSON, P. D. SEYMOUR, AND ROBIN THOMAS Description. We use the algorithm of Hopcroft and Karp =-=[4]-=- to find a perfect matching M in G. If G has no perfect matching, then every orientation of G is Pfaffian. In that case we output an arbitrary orientation of G and stop. Otherwise we use 9.1 to delete... |

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Citation Context .... If G is a graph and X ⊆ V (G), we denote by NG(X) the set of vertices of V (G) − X adjacent to a vertex in X. We need the following well-known characterization of k-extendable bipartite graphs (see =-=[13]-=-). 3.1. Let G be a connected bipartite graph with bipartition (A,B), and let k ≥ 0 be an integer. Then the following two conditions are equivalent. (i) G is k-extendable, and (ii) |A| = |B|, and for e... |

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Citation Context ...does not have a Pfaffian orientation by 7.3. We output that information and stop. Thus we may assume that G has at most 2n − 4 edges. For every u,v ∈ V (G) we use the algorithm of Hopcroft and Tarjan =-=[5]-=- to find all trisectors X of G with u,v ∈ X. Thus we find all trisectors of G in time O(n 3 ), and apply 9.2. Let G be a connected 1-extendable bipartite graph. Let C0 = {G}, and assume that the sets ... |

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Citation Context ...hen G contains K3,3 by 5.5 (delete 1-19, 3-19 and 17-20).960 NEIL ROBERTSON, P. D. SEYMOUR, AND ROBIN THOMAS 6. Proof of the main result We start with six lemmas. The first is a theorem of Kasteleyn =-=[6]-=-. 6.1. Every planar graph admits a Pfaffian orientation. 6.2. The graph K3,3 does not admit a Pfaffian orientation. Proof. Let (A,B) be a bipartition of K3,3, and let C be the set of all circuits of K... |

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Citation Context ...n this case we use a linear planarity algorithm such as [26] to either find a planar drawing of G, or determine that G is nonplanar. If we find a planar drawing of G we use Kasteleyn’s algorithm [6], =-=[7]-=- (see also972 NEIL ROBERTSON, P. D. SEYMOUR, AND ROBIN THOMAS [13]) to output a Pfaffian orientation of G and stop. If G is nonplanar, then we check if G is isomorphic to the Heawood graph. If it is,... |

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Citation Context ... a certain “obstruction” submatrix of A whose presence implies that A has no Pólya matrix. The algorithm easily extends to matrices with nonnegative entries, as pointed out by Vazirani and Yannakakis =-=[25]-=-. Our results are best stated and proved in terms of bipartite graphs. By a graph we mean a finite simple undirected graph, that is, one with no loops or parallel edges. A set M of edges of G is a mat... |

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Citation Context ...5, then we output the statement that G has no Pfaffian orientation, and stop. By 8.9 this statement is correct. We now assume that L is empty. In this case we use a linear planarity algorithm such as =-=[26]-=- to either find a planar drawing of G, or determine that G is nonplanar. If we find a planar drawing of G we use Kasteleyn’s algorithm [6], [7] (see also972 NEIL ROBERTSON, P. D. SEYMOUR, AND ROBIN T... |

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Citation Context ...roblem for directed graphs (see [19], [20], [21], [22], [23]), by [11], [18] it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by =-=[8]-=- it solves the problem of determining which real n × n matrices are sign-nonsingular.932 NEIL ROBERTSON, P. D. SEYMOUR, AND ROBIN THOMAS The Heawood graph Figure 1 See also [1] for variations of sign... |

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Citation Context ...this also gives a polynomialtime algorithm to decide if a 1-1 square matrix has a Pólya matrix, by [25] this solves the even circuit problem for directed graphs (see [19], [20], [21], [22], [23]), by =-=[11]-=-, [18] it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by [8] it solves the problem of determining which real n × n matrices are... |

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Citation Context ...faffian orientation. By 1.1 this also gives a polynomialtime algorithm to decide if a 1-1 square matrix has a Pólya matrix, by [25] this solves the even circuit problem for directed graphs (see [19], =-=[20]-=-, [21], [22], [23]), by [11], [18] it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by [8] it solves the problem of determining w... |

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Citation Context ... the following. 1.1. Let A be a 0-1 square matrix, and let G be the associated bipartite graph. Then A has a Pólya matrix if and only if G has a Pfaffian orientation.PFAFFIAN ORIENTATIONS 931 Little =-=[9]-=- proved the following elegant characterization of bipartite graphs that admit a Pfaffian orientation (and hence of matrices that admit a Pólya matrix). We say that a graph G is a subdivision of a grap... |

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Citation Context ...as a Pfaffian orientation. By 1.1 this also gives a polynomialtime algorithm to decide if a 1-1 square matrix has a Pólya matrix, by [25] this solves the even circuit problem for directed graphs (see =-=[19]-=-, [20], [21], [22], [23]), by [11], [18] it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by [8] it solves the problem of determi... |

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Citation Context ...lso gives a polynomialtime algorithm to decide if a 1-1 square matrix has a Pólya matrix, by [25] this solves the even circuit problem for directed graphs (see [19], [20], [21], [22], [23]), by [11], =-=[18]-=- it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by [8] it solves the problem of determining which real n × n matrices are sign-... |

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Citation Context ...96. In the fall of that year, William McCuaig informed us that he had already obtained the same result, but had not publicly announced it. Consequently, a joint announcement of the result was made in =-=[16]-=-; but since the proof methods are quite different, we agreed with McCuaig that both papers should be submitted for publication independently. We would like to thank an anonymous referee for carefully ... |

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Citation Context ...ut are adjacent to a vertex of G\V (G′ i ), and let Gi be obtained from G ′ i by joining each vertex of Y ′ i by an edge to ui. We say that G is a 2-sum of G1 and G2. The following lemma follows from =-=[10]-=-. 7.1. Let G1 and G2 be bipartite graphs, let i ∈ {0,2}, and let G be an i-sum of G1 and G2. Then G has a Pfaffian orientation if and only if both G1 and G2 have Pfaffian orientations. We deduce the f... |

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Citation Context ...n orientation. By 1.1 this also gives a polynomialtime algorithm to decide if a 1-1 square matrix has a Pólya matrix, by [25] this solves the even circuit problem for directed graphs (see [19], [20], =-=[21]-=-, [22], [23]), by [11], [18] it solves the problem of determining which hypergraphs with n vertices and n hyperedges are minimally nonbipartite, and by [8] it solves the problem of determining which r... |

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1 | Even cycles and H-homeomorphisms - Galluccio, Loebl |