Combinatorial methods for constructing bipartite uniform minimal quasicomplete graphs (symmetrical block designs)

On the route-oriented method of system diagnostics in digital systems structured as a symmetric bipartite graph

“…One of the advantages of this structure when it operates correctly [1] is that any pair of vertices from subset X is related via a single vertex from subset Y . This is exactly the vertex from subset Y that appears as a suspect when two vertices from subset X fail at the same time.…”

“…3) fill with dashes (-) the cells in rows A 1 , A Z ; 4) then write in the cells of these rows diagnostic results according to the following rules: a) write 1 to cells on the intersections of row A 1 and columns titled with symbols of vertices u3, u8, u1 since cells of these columns located above contain at least one 1 and do not contain any zeros; b) write Z to cells on the intersections of row A Z and 7 columns titled with symbols of arcs l(j−i) since at least one of the vertices (j or i) that each of these arcs connect is suspected to be faulty (there is a 1 in the corresponding cell of row A 1 ); 5) deciphering the resulting records in rows A 1 and A Z , we get the following diagnostic conclusions: a) according to the locations of ones in row A 1 , processors 1 and 3 and memory unit 8 are suspected to be faulty, even though the original fault situation included only vertices 1 and 3; vertex 8 is suspected to be faulty since this triple fault (1,8,3) is indistinguishable from the original double fault (1, 3), as we have shown in Section 3; b) according to the locations of Z symbols in row A Z , state of the communication lines (13-3), (14-3), (8-3), (8-2), (8-1), (9-1), and (10-1) remains unknown since their manifestations are "masked" by the state of adjacent vertices 1, 3, 8 that are suspected to be faulty.…”

“…Rows 4, 5 show potential syndromes for double and triple faults of some of the components 1, 8, 3. Finally, rows 6-11 show potential syndromes for a number of faults, including the faults of communication lines (1)(2)(3)(4)(5)(6)(7)(8), .…”

“…When the DM signals to begin the control initial unit 1 will perform the following actions: -form an intermediate (Table 3) the row T 1 , find in it indices (1,9,4), (1,8,2), (1, 10, 6) of tested subsystems U 1,9,4 , U 1,8,2 , U 1,10,6 , and transmit to unit 4 (2, 6) the following command: "Perform self-testing of adjacent subsystems U 1,9,4 , (U 1,8,2 , U 1,10,6 ) and transmit test results into unit 1." (1-10), 10, (10-6), 6 T 4 4 (4-11), 11, (11-6), 6 (4-9), 9, (9-5), 5…”

“…A symmetric bipartite graph is a graph consisting of two subsets of vertices, X and Y , of equal size n with edges one of whose end vertices belongs to the subset X(Y ), and the other, to subset Y (X) [1]. Among possible fields of application for such graphs one can list structures of multiprocessor computer systems (MCS), where, for instance, the subset of vertices X represents the collection of processor elements or computational machines, and the subset of vertices Y represents units of memory banks.…”