# Linked Coupling Faults Detection by Multirun March Tests
## 2 Linked Coupling Faults
If a \(CF(a_{i},a_{j})\) is included in an \(LCF\) consisting of \(\mathbf{r}\) cells, then the number of roles for one of the cells \(a_{i}\) or \(a_{j}\) forming the \(\mathrm{CF}\) must be at least two, and the total number for both cells of the \(\mathrm{CF}\) lies in the range from \(3\) to \(4r-6\). The validity of this statement follows from the fact that if each of the cells \(a_{i}\) and \(a_{j}\) forming a single \(CF\) has only one role, then this \(CF\) is not included in the \(LCF\). Such a fault is not associated with other \(CF\)s because the cells included in this fault do not have roles that connect it with other memory cells or other \(CF\)s. Thus, the presence of at least one of the cells \(a_{i}\) and \(a_{j}\) of the \(CF\) of more than one role provides a connection between other single \(CF\)s, including in the \(LCF\). Thus, the minimum number of roles for a \(CF\) in an \(LCF\) is three. The upper bound of the number of roles for the fault cells of a \(CF\) participating in the \(LCF\) (equal to \(4r-6\)) is achieved when each of the two fault cells plays the roles of both the aggressor victim in relation to the other \(r-2\) cells. Two more roles are possible between the \(CF\) cells \(a_{i}\) and \(a_{j}\). Then, the maximum number of roles of \(CF\) cells included in the \(LCF\) is defined as \(2(r-2)+2(r-2)+2=4r-6\). The minimum number (\(\min(n)\)) of single \(CF\)s that form an \(LCF\) is \(r-1\). This result follows from Statement 1, corresponding to the minimum number of roles for each \(CF\), two describing the \(CF\) itself, and the third defining the relationship with another \(CF\). An example would be when, in each subsequent \(CF\), the aggressor cell is also the victim, for which the victim cell of the current fault acts as the aggressor. Thus, \(\min(n)=r-1\), which is also achievable for the second extreme case, where one of the \(r\)\(LCF\) cells is the aggressor for the remaining \(r-1\) cells. We also have \(r-1\) single \(CF\)s, where one of the cells performs more than two roles, in this case, \(r-1\) roles. Examples of such faults \(LCF3\) and \(LCF4\) for \(r=3\) are depicted in Figure 2. The maximum number (\(\max(n)\)) of single \(CF\)s generating an \(LCF\) is achieved when any possible pair of cells from \(r\) cells forms two \(CF\)s, respectively, and the first cell plays the role of \(A\), the second \(V\), and vice versa. Thus, \(\max(n)=2\cdot\left( \begin{array}{c}r\\ 2\end{array} \right)=r(r-1)\). An example of an \(LCF\) consisting of the maximum number of single \(CF\)s for \(r=3\) is presented in Figure 3 as the \(LCF5\). Thus, for an \(LCF\) consisting of \(r\) cells, the number \(n\) of single \(CF\)s forming this fault is in the following range:  
\[min(n)=r-1\leq n\leq max(n)=r(r-1). \tag{2}\]  
The resulting limit values for the number of \(CF\)s included in the \(LCF\) correspond to similar values for the number of edges in connected labeled directed graphs consisting of \(r\) vertices. Using graph theory, the number of types of \(LCF\)s, consisting of \(r\) memory cells, can be estimated as the total number of simple labeled directed graphs with \(r\) vertices. This  
Figure 3: Linked coupling faults for \(r=3\). number \(2r(r-1)\) grows exponentially with the number of cells \(r\) and can be used as an upper bound for the number of \(LCF\) varieties. The min(\(n\)) and max(\(n\)) values of the number of single \(CF\)s that form an \(LCF\) (2), including \(r\) memory cells, were obtained without considering the type of CFs, their location in the memory, and the specifics of their mutual influence on each other.