Points articulation

Fig. 2. Some of the diagrams occurring in Mt. Articulation points are indicated by arrows, (a) and (b) are irreducible diagrams (c) and (d) are linked reducible and unlinked respectively.

We study next the dynamical irreducibility condition which appeared in the definition of the transport operator. It eliminates from this quantity the reducible collision processes where the particles coming from infinity interact, recede to an infinite distance from one another, and then interact again. We define an extended transport operator from which the irreducibility condition is eliminated and which involves this time the reducible collisions. The relation between the transport operator and the extended transport operator is made explicit by means of a correspondence between the dynamical processes and the Mayer graphs for equilibrium. In this respect, we demonstrate, in these graphs, the importance of the role of the articulation points. 

To this end we shall associate with each contribution of Eq. (70) a connected graph30 constructed in the following fashion each particle is represented by a point and we connect two points by one line when the two particles considered interact one or more times in the contribution in question. In a graph there may exist an articulation point at which time the graph can be divided... 

Let us consider first a contribution such that the corresponding graph has no articulation point (see Fig. 6). Any intermediate... 

Let us see now what happens when one has an articulation point (see Fig. 7a). This time the intermediate state i defines... 

The irreducible contributions to generated by the two graphs with an articulation point are included in VF(423), which also contains all the contributions coming from the graph with three lines. The factor 1/— z in Eq. (74) comes from a propagator (66) with k = 0. 

We now have to apply the same methods to

more than four lines) and furnishes only irreducible terms (included in 1P(4234)). On the other hand, graph (b) contains reducible contributions. To obtain them, we again suppress in Eq. (70) the 8Lm corresponding to the... 

To do this we must study the structure of the operator <0 B( 34> 0) given by expression (25"). This appears as a sum of products of the operators ) preceded from the left by an operator 012. It is necessary to remark that for each of the terms and for every pair of operators S( l), never more than one particle is repeated. These particles common to two S(IT- ) will be articulation points in the graph language defined above and, consequently, all the intermediate states between the operators ) will be k = 0. Furthermore, this remark is general and valid to all orders in the concentration in Cohen s formalism. 

The concepts "hierarchy", "articulation points", "chains" and "antichains" are very basic and simple ones, which direct into the structural analysis of digraphs. These concepts will be explained in the next section. 

Long chains, hierarchies and articulation points indicate specific data structures. The role of hierarchies will be explained by a two dimensional scheme (Fig. 13) Several objects may be located as points within the two rectangles Hi and H2. Comparing one object of Ht with one of H2 will lead to qi (of x g Hi) > qi (of y g H2), whereas q2 (of x g Ht) < q2 (of y g H2). Hence no object of H is comparable with that of H2. In Neggers Kim 1998 a rather nice wording is found for the objects belonging to the field F and P These are the future objects relative to the objects in the field P, which are called the objects in the past. 

An articulation point in a connected graph is a point whose removal breaks a graph into two or more unconnected parts such that at least one part contains no root point and at least one field point. (A slightly more general definition, which we will not need, is required to define an articulation point in a disconnected diagram.) This definition holds even when the graph has no root point or one root point. See Fig. 2 for illustrations of this definition. 

A graph is irreducible if it has no articulation points. It follows from these definitions that a graph with no root points is irreducible if and only if it is at least doubly connected. Also, a graph with two root points is irreducible if and only if it is at least doubly connected or it would become at least doubly connected when a bond between the roots is drawn. 

A pair of reducible points in an irreducible graph is a pair of points that are connected by a bond and/or that are a pair of articulation points. When a pair of reducible points is removed from a diagram, the diagram becomes disconnected into two or more parts. Some parts may simply be bonds with no point on each end, if the pair of reducible points were connected by one or more bonds. Some parts may be collections of field points connected by bonds and containing some bonds with no point on one end, if the pair of reducible points were a pair of articulation points. Some parts may be collections of field points and root points, containing some bonds with no point at one end, if the original... 

Fig. 2. Illustration of the definition of an articulation point. In the two graphs at the top, the points indicated by stars are articulation points. When these points are removed (leading to the structures drawn immediately below the graphs), the graph becomes disconnected and at least one of the disconnected parts has no root point and one or more field points. (The symmetry numbers of the two graphs at the top are 4 and 1.)...

Figure 19 The active side chains are shown in (a). How the side chains are divided into biconnected components is illustrated in (b). Each side chain is numbered with a DEN (circle) and a low number (square next to corresponding residue). A difference between the DEN and the low number indicates the existence of an articulation point for the group of active sidechains. This image was adapted from Canutescu et al. ...

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