Pfaffian Orientations Research Articles

A Polynomial Time Algorithm for Recognizing Near-Bipartite Pfaffian Graphs

Abstract A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. A single ear of a connected graph G is a path P of odd length in G whose internal vertices, if any, have degree two in G. Let G – P denote the graph obtained from G by deleting the edges and internal vertices of G. A double ear of G is a pair ( R 1 , R 2 ) of vertex-disjoint single ears of G. A matching covered graph G is near-bipartite if it is non-bipartite and there is a double ear ( R 1 , R 2 ) such that G – R 1 – R 2 is matching covered and bipartite, but G – R i is not matching covered, for i = 1 , 2 . In 2000, C. Little and I. Fischer characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [Little, C.H.C. and I. Fischer, A characterisation of Pfaffian near bipartite graphs, J. Combin. Theory Ser. B 82 (2001), pp. 175–222]. However, their characterization does not imply a polynomial time algorithm to determine whether a near-bipartite graph is Pfaffian. In this report, we give such an algorithm. Our algorithm is based in an Inclusion-Exclusion principle and uses as a subroutine an algorithm by McCuaig [McCuaig, W., Polya's permanent problem, The Electronic J. of Combin. 11 (2004)] and, independently, by Robertson, Seymour and Thomas [Robertson, N., P.D. Seymour and R. Thomas, Permanents, Pfaffian orientations and even directed circuits, Ann. of Math. (2) 150 (1999), pp. 929–975] that determines whether a bipartite graph is Pfaffian.

Pfaffian labelings and signs of edge colorings

We relate signs of edge-colorings (as in classical Penrose’s result) with “Pfaffian labelings”, a generalization of Pfaffian orientations, whereby edges are labeled by elements of an Abelian group with an element of order two. In particular, we prove a conjecture of Goddyn that all k-edge-colorings of a k-regular Pfaffian graph G have the same sign. We characterize graphs that admit a Pfaffian labeling in terms of bricks and braces in their matching decomposition and in terms of their drawings in the projective plane.

On the number of dissimilar pfaffian orientations of graphs

A subgraph H of a graph G is conformal if G - V(H) has a perfect matching. An orientation D of G is Pfaffian if, for every conformal even circuit C , the number of edges of C whose directions in D agree with any prescribed sense of orientation of C is odd. A graph is Pfaffian if it has a Pfaffian orientation. Not every graph is Pfaffian. However, if G has a Pfaffian orientation D , then the determinant of the adjacency matrix of D is the square of the number of perfect matchings of G . (See the book by Lovasz and Plummer [Matching Theory . Annals of Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.] A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. The study of Pfaffian orientations of graphs can be naturally reduced to matching covered graphs. The properties of matching covered graphs are thus helpful in understanding Pfaffian orientations of graphs. For example, say that two orientations of a graph are similar if one can be obtained from the other by reversing the orientations of all the edges in a cut of the graph. Using one of the theorems we proved in [M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, Optimal ear decompositions of matching covered graphs. J. Combinat. Theory B 85 (2002) 59–93] concerning optimal ear decompositions, we show that if a matching covered graph is Pfaffian then the number of dissimilar Pfaffian orientations of G is 2b(G) , where b(G) is the number of “bricks” of G . In particular, any two Pfaffian orientations of a bipartite graph are similar. We deduce that the problem of determining whether or not a graph is Pfaffian is as difficult as the problem of determining whether or not a given orientation is Pfaffian, a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs. Discrete Appl. Math. 25 (1989) 179–180]. We establish a simple property of minimal graphs without a Pfaffian orientation and use it to give an alternative proof of the characterization of Pfaffian bipartite graphs due to Little [ A characterization of convertible (0,1) -matrices. J. Combinat. Theory B 18 (1975) 187–208] .

On the hardness of efficiently approximating maximal non- L submatrices

The sign pattern of a real matrix A is the matrix obtained by replacing each entry of A by its sign. A real matrix A is an L- matrix if every real matrix with the same sign pattern as A has linearly independent columns. L-matrices arise naturally in and are essential to the study of sign-solvability and related notions. In special cases, the L-matrix property has connections to the even dicycle problem, Pfaffian orientations, and Pólya’s permanent problem. Unfortunately, the problem of recognizing L-matrices is known to be co- NP -complete in general. We elaborate in this vein by showing a polynomial-time inapproximability result for Max-not- L -rows, a particular optimization version of L-matrix recognition, by means of an approximation-preserving reduction from the previously studied problem Max-not-equal-2-Sat.

Permanents, Pfaffian Orientations, and Even Directed Circuits

Given a 0-1 square matrix A, when can some of the 1’s be changed toi1’s in such a way that the permanent of A equals the determinant of the modifled 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 nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a \Pfa‐an 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 of the above problems are equivalent. We prove a structural characterization 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 Pfa‐an orientation if and only if it can be obtained by piecing together (in a specifled way) planar bipartite graphs and one sporadic nonplanar bipartite graph.

On the Theory of Pfaffian Orientations. II. $T$-joins, $k$-cuts, and Duality of Enumeration

This is a continuation of our paper "A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of $T$-joins and $k$-cuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of an edge-cut is polynomially solvable for the instances $(G,w)$ where $G$ is a graph embedded on an arbitrary fixed orientable surface and the weight function $w$ has only a bounded number of different values. We also survey the related results concerning a duality of the Tutte polynomial, and present an application for the weight enumerator of a binary code. In a continuation of this paper which is in preparation we present an application to the Ising problem of three-dimensional crystal structures.

Pfaffian orientations, 0–1 permanents, and even cycles in directed graphs

The following issues in computational complexity remain imprecisely understood: The striking difference in the complexities of computing the permanent and determinant of a matrix despite their similar looking formulae, the complexity of checking if a directed graph contains an even length cycle, and the complexity of computing the number of perfect matchings in a graph using Pfaffian orientations. Via polynomial time equivalences, we show inter-relationships among these issues.