Graphs with table constraints on reachability

On directed graphs defined a new kind of reachability restriction – table constraints on reachability. Each edge of the graph assigned a certain element of the monoid. Some element of the monoid selected and fixed. This element does not equal to the unit of the monoid. This element is called forbidden. Each way on the graph is associated with characteristic – a vector, whose length is equal to the number of way edges. First element of the characteristic is equal to the element of the monoid which corresponded to the first edge of the way. Each successive element of the characteristic is equal to the previous element of the characteristics plus the value of an element of the monoid, corresponding to the next edge of the way. A way on the graph is considered as valid if the way characteristic does not contain a forbidden element. The construction of the scan-graph, which is built on the original graph, is described. The transition to the scan-graph allows you to solve the problems of the reachability on such graphs, shortest ways and random walks on the graphs with table constraints on reachability.