A note on the disjunctive index of joined a-perfect graphs

In this paper we present a lower bound of the disjunctive rank of the facets describing the stable set polytope of joined a-perfect graphs. This class contains near-bipartite, t-perfect, h-perfect and complement of fuzzy interval graphs, among others. The stable set polytope of joined a-perfect graphs is described by means of full rank constraints of its node induced prime antiwebs. As a first step, we completely determine the disjunctive rank of all these constraints. Using this result we obtain a lower bound of the disjunctive index of joined a-perfect graphs and prove that this bound can be achieved. In addition, we completely determine the disjunctive index of every antiweb and observe that it does not always coincide with the disjunctive rank of its full rank constraint.     

Bond graph model-based system mode identification and mode-dependent fault thresholds for hybrid systems

Hybrid system models exploit the modelling abstraction that fast state transitions take place instantaneously so that they encompass discrete events and the continuous time behaviour for the while of a system mode. If a system is in a certain mode, e.g. two rigid bodies stick together, then residuals of analytical redundancy relations (ARRs) within certain small bounds indicate that the system is healthy. An unobserved mode change, however, invalidates the current model for the dynamic behaviour. As a result, ARR residuals may exceed current thresholds indicating faults in system components that have not happened. The paper shows that ARR residuals derived from a bond graph cannot only serve as fault indicators but may also be used for bond graph model-based system mode identification. ARR residuals are numerically computed in an off-line simulation by coupling a bond graph of the faulty system to a non-faulty system bond graph through residual sinks. In real-time simulation, the faulty system model is to be replaced by measurements from the real system. As parameter values are uncertain, it is important to determine adaptive ARR thresholds that, given uncertain parameters, allow to decide whether the dynamic behaviour in a current system mode is the one of the healthy system so that false alarms or overlooking of true faults can be avoided. The paper shows how incremental bond graphs can be used to determine adaptive mode-dependent ARR thresholds for switched linear time-invariant systems with uncertain parameters in order to support robust fault detection. Bond graph-based hybrid system mode identification as well as the determination of adaptive fault thresholds is illustrated by application to a power electronic system easy to survey. Some simulation results have been analytically validated.     

组合计数的群论与计算机方法

本文综述组合数学和图论中解决计数问题的群论与计算机方法及其最新发展。传统的计数方法得到有限的计数公式或递推公式等，然而许多复杂的问题很难得到有限的表达式，即使能得到，公式也往往非常复杂。由于计算机技术的发展不仅使复杂的计数公式有了实际意义，而且可以设计恰当的计算方法进行数值计算，使计数问题有更为广阔的发展领域。另一方面，为了计算不同构的图或组合结构，最有效的方法是群论方法，因此把群论方法与计算机方     

Edge-forwarding index of star graphs and other Cayley graphs

We present a technique for building, in some Cayley graphs, a routing for which the load of every edge is almost the same. This technique enables us to find the edge-forwarding index of star graphs and complete-transposition graphs.     

Partitioned Tensor Products and Their Spectra

A pleasant family of graphs defined by Godsil and McKay is shown to have easily computed eigenvalues in many cases.     

Sources and Sinks in Comparability Graphs

We prove that a subset S of vertices of a comparability graph G is a source set if and only if each vertex of S is a source and there is no odd induced path in G between two vertices of S. We also characterize pairs of subsets corresponding to sources and sinks, respectively. Finally, an application to interval graphs is obtained.     

Two characterisations of the minimal triangulations of permutation graphs

A minimal triangulation of a graph is a chordal supergraph with an inclusion-minimal edge set. Minimal triangulations are obtained from adding edges only to minimal separators, completing minimal separators into cliques. Permutation graphs are the comparability graphs whose complements are also comparability graphs. Permutation graphs can be characterised as the intersection graphs of specially arranged line segments in the plane, which is called a permutation diagram. The minimal triangulations of permutation graphs are known to be interval graphs, and they can be obtained from permutation diagrams by applying a geometric operation, that corresponds to the completion of separators into cliques. We precisely specify this geometric completion process to obtain minimal triangulations, and we completely characterise those interval graphs that are minimal triangulations of permutation graphs.     

Snarks and Flow-Critical Graphs

It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutteʼs Flow Conjectures.