Structural characterization of graphs with complementary zero forcing number at most three

Provide a structural characterization of all graphs with complementary zero forcing number mz(G) at most 3 by describing precisely the graphs that contain none of the 25 forbidden graphs shown in Figure 1 as an induced subgraph.

Background

The paper defines mz(G)=|V(G)|−Z(G), where Z(G) is the zero forcing number, and proves that graphs with mz(G) at most 3 are exactly those avoiding a finite family of 25 minimal forbidden induced subgraphs. Although the forbidden family is determined computationally, the authors state that a structural description of all graphs avoiding these configurations is still the outstanding problem motivating the subsequent partial results.

The remainder of the paper obtains complete descriptions in some restricted cases, including graphs of girth at least 4, and develops partial structural and computational analyses for graphs of girth 3. These results do not constitute a complete structural characterization in the general case.