Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.     

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.     

Generating random regular graphs

An algorithm is described which generates a random labeled cubic graph on n vertices. Also described is a procedure which, if successful, generates a random (0,1)-matrix with prescribed row and column sums. The latter yields procedures which, if successful, generate random labeled graphs with specified degree sequence and random labeled bipartite graphs with specified degree sequences. These procedures can be implemented so that each trial requires time which is linear in the number of vertices plus edges, but in generating a random r-regular graph, the probability of success of a given trial is about $\text{exp}\text{(}\text{(1 ? r}{}^2\text{)}\text{4}\text{)}$, which is prohibitively small for large r. Comparisons are made between the complexities of the two methods of generating random cubic graphs. The two general schemes presented derive from methods which have been used to enumerate regular graphs, both asymptotically and exactly.     

NP‐completeness of list coloring and precoloring extension on the edges of planar graphs

In the edge precoloring extension problem, we are given a graph with some of the edges having preassigned colors and it has to be decided whether this coloring can be extended to a proper k‐edge‐coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP‐complete on (a) planar 3‐regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series‐parallel graphs. This improves previous results of Easton and Parker 6 , and Fiala 8 . © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 313–324, 2005     

Geodetic orientations of complete k-partite graphs

Ore defined a graph to be geodetic if and only if there is a unique shortest path between two points, and posed the problem of characterizing such graphs. Here this problem is studied in the context of oriented graphs and such geodetic orientations are characterized first for complete graphs (geodetic tournaments), then for complete bipartite and complete tripartite graphs, and finally for complete k-partite graphs.     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

Dimers, tilings and trees

Generalizing results of Temperley (London Mathematical Society Lecture Notes Series 13 (1974) 202), Brooks et al. (Duke Math. J. 7 (1940) 312) and others (Electron. J. Combin. 7 (2000); Israel J. Math. 105 (1998) 61) we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to “discrete analytic functions” on the bipartite graph.The equivalence is extended to infinite periodic graphs, and we classify the resulting “almost periodic” tilings and harmonic functions.     

导出匹配问题的NP-完全性以及导出匹配可扩问题的CO-NP-完全性

图Ｇ的一个匹配Ｍ是导出的,若Ｍ是图Ｇ的一个导出子图。图Ｇ是导邮匹配可扩的（简记ＩＭ－可扩的）,若图Ｇ的任一导出匹配均含于图Ｇ的一个完美匹配当中。本文我们将证明如下结果。⑴对无爪图而言,问题“给定图Ｇ以及一个正整数ｒ,确定是否存在图Ｇ的一个导出匹配Ｍ使得Ｍ≥ｒ”是ＮＰ－完全的。⑵对直径为２的图以及直径为３的偶图,问题“确定一个给定图是否为导出匹配可扩的”是ＣＯ－ＮＰ完全的;而对完全多部图而言,问题“     

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang–Swendsen–Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when kΔ/2, provided k is larger than some absolute constant k₀.     

Constructing cospectral graphs via a new form of graph product

We present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.     

The number of bipartite tournaments with a unique given factor

We give a recursive function in order to calculate the number of all non-isomorphic bipartite tournaments containing an unique hamiltonian cycle. Using this result we determine the number of all nonisomorphic bipartite tournaments that admit an unique factor isomorphic to a given 1-diregu-lar bipartite graph.     

An Approximate Version of Sidorenko’s Conjecture

A beautiful conjecture of Erdős-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.     

The shifted Turán sieve method on tournaments II

In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which contain a small number of r-cycles. In this paper, we improve our sieve inequality and apply it to obtain an upper bound for the number of bipartite tournaments which contain a number of 2r-cycles far from the average. We also provide the exact bound for the number of tournaments which contain few 3-cycles, using other combinatorial arguments.     

Sampling Eulerian orientations of triangular lattice graphs

We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although Mihail and Winkler (1989) showed that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist non-solid subgraphs (subgraphs with holes) for which the chain will always take an exponential amount of time to converge. Finally, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs (Mihail and Winkler had already established this for general graphs).     

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

d , and the testing algorithm can perform queries of the form: ``who is the ith neighbor of vertex v'. The tester should determine with high probability whether the graph is bipartite or ε-far from bipartite for any given distance parameter ε. Distance between graphs is defined to be the fraction of entries on which the graphs differ in their incidence-lists representation. Our testing algorithm has query complexity and running time where N is the number of graph vertices. It was shown before that queries are necessary (for constant ε), and hence the performance of our algorithm is tight (in its dependence on N), up to polylogarithmic factors. In our analysis we use techniques that were previously applied to prove fast convergence of random walks on expander graphs. Here we use the contrapositive statement by which slow convergence implies small cuts in the graph, and further show that these cuts have certain additional properties. This implication is applied in showing that for any graph, the graph vertices can be divided into disjoint subsets such that: (1) the total number of edges between the different subsets is small; and (2) each subset itself exhibits a certain mixing property that is useful in our analysis. Received: February 6, 1998     

两类广义控制问题的NP-完全性

研究两类广义控制问题的复杂性: k-步长控制问题和k-距离控制问题, 证明了k-步长控制问题在弦图和平面二部图上都是NP-完全的. 作为上述结果的推论, 给出了k-距离控制问题在弦图和二部图上NP-完全性的新的证明, 并进一步证明了k-距离控制问题在平面二部图上也是NP-完全的.     

Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.