On the problem of consistent marking of a graph

AbstractA marked graph is a graph in which each vertex is given a sign + or −. We call such a graph consistent if every cycle has an even number of — signs. Consistent marked graphs arise in the study of communication networks and social networks. We discuss the problem of characterizing graphs that can be consistently marked using at least one — sign, reduce the problem to blocks, and solve it for blocks whose longest cycle has length at most 5