The zero forcing number of a graph is the minimum number of vertices
in a zero forcing set of . It is a graph parameter
that is closely related to the maximum nullity and that bounds it from above:
(AIM 2008, Barrett et al. 2025).
Results from the American Institute of Mathematics (2008) showed that for any graph on 6 or fewer
vertices. DeLoss et al. (2008) subsequently
computed
and bounds on
for all 7-vertexgraphs,
and the resulting table establishes that for all graphs on 7 vertices
(Barrett et al. 2025). Barrett et al. (2025) subsequently showed that
for all graphs on 8 vertices
with the exception of a set of 7 graphs
denoted
to
and which are illustrated above. As it turns out, is the graph complement
of the 3-Plummer-Toft graph.
For the generalized Petersen graphs with and , the general upper bound
is
(Alameda et al. 2018, p. 58). Previously known exact values include the
prism graph values and for (Alameda et al. 2018, p. 58). Alameda et
al. (2018, p. 60) also proved for every integer. Rashidi et al. (2020, pp. 185 and
189-190) proved
for
and
for . Krishnan et al. (2026) give the exact
values for all members of the generalized
Petersen graph subfamilies and : for every and for every . The smaller values of
for , 8, ..., 12 are 6, 6, 6, 8, 7, 7, and those of for , 10, ..., 17 are 6, 6, 7, 6, 8, 8, 9, 8, 9. Krishnan
et al. (2026) report AI assistance with computational code and Lean development
for the
stabilization theorem.
For maximal biconnectedouterplanar graphs, Han and Liang (2026) report an exact formula and a linear-time algorithm
that constructs a minimum zero forcing set and
its forcing chronology from indexed triangle adjacency. Their formula accounts for
obstructions to orienting a minimum path cover into a valid forcing process. Han
and Liang (2026) explicitly state that the AI-assisted mathematical derivations and
computations are still being checked and rewritten. These reported results remain
provisional.
Abiad, A.; Flagg, M.; Ghasemi Nezhad, S.; and Zeijlemaker, S. "The Zero Forcing Number of Graph Powers." 2 Oct 2026. https://arxiv.org/abs/2610.03892.AIM
Minimum Rank--Special Graphs Work Group. "Zero Forcing Sets and the Minimum
Rank of Graphs." Lin. Alg. Appl.428, 1628-1648, 2008.Alameda,
J. S.; Curl, E.; Grez, A.; Hogben, L.; Kingston, O.; Schulte, A.; Young, D.;
and Young, M. "Families of Graphs with Maximum Nullity Equal to Zero Forcing
Number." Special Matrices6, 56-67, 2018. https://doi.org/10.1515/spma-2018-0006.Almodovar,
E.; DeLoss, L.; Hogben, L.; Hogenson, K.; Murphy, K.; Peters, T.; and Ramírez,
C. A. "Minimum Rank, Maximum Nullity and Zero Forcing Number for Selected Graph
Families." Involve: A Journal of Mathematics3, 371-392, 2010.American
Institute of Mathematics. "Graph Catalog: Families of Graphs." https://aimath.org/WWN/matrixspectrum/catalog2.html.American
Institute of Mathematics. "AIM Minimum Rank Graph Catalog." http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/super.American
Institute of Mathematics. "AIM Minimum Rank Graph Catalog References."
https://web.archive.org/web/20160207084159/http://orion.math.iastate.edu/lhogben/AIMmrgraphcatrefs.pdf.Barrett,
W.; Hunnell, M.; Hutchens, J.; and Sinkovic, J. "The Classification of Graphs
on 8 Vertices with Coinciding Zero Forcing Number and Maximum Nullity." 12 Jun
2025. https://arxiv.org/abs/2506.10726.Barrett,
W.; van der Holst, H.; and Loewy, R. "Graphs Whose Minimal Rank Is Two."
Elec. J. Lin. Alg.11, 258-280, 2004.DeLoss, L.; Grout,
J.; Hogben, L.; McKay, T.; Smith, J.; and Tims, J. "Table of Minimum Ranks of
Graphs of Order at Most 7 and Selected Optimal Matrices." 4 Dec 2008. https://arxiv.org/abs/0812.0870.Fallat,
S. M. and Hogben, L. "Minimum Rank, Maximum Nullity, and Zero Forcing Number
of Graphs." Ch. 46 in Handbook of Linear Algebra, 2nd ed. Boca Raton,
FL: CRC Press, 2014.Han, Z. and Liang, Y. "Zero Forcing on Maximal
Outerplanar Graphs: Capacities and Phase Repair." 22 Oct 2026. https://arxiv.org/abs/2610.31138.Krishnan,
A.; Ma, H.; and Zhang, W. "Sharp Zero Forcing Numbers for the Generalized Petersen
Graphs
and ."
16 Oct 2026. https://arxiv.org/abs/2610.24059.Nylen,
P. M. "Minimum-Rank Matrices With Prescribed Graph." Lin. Alg.
Appl.248, 303-316, 1996.Rashidi, S.; Shajareh Poursalavati,
N.; and Tavakkoli, M. "Computing the Zero Forcing Number for Generalized Petersen
Graphs." J. Algebra Combin. Discrete Struct. Appl.7, 183-193,
2020. https://doi.org/10.13069/jacodesmath.729465.
of a graph 
is the minimum number of vertices
in a zero forcing set of 
. It is a graph parameter
that is closely related to the maximum nullity 
and that bounds it from above:
for any graph on 6 or fewer
vertices. DeLoss et al. (2008) subsequently
computed 
and bounds on 
for all 7-vertex graphs,
and the resulting table establishes that 
for all graphs on 7 vertices
(Barrett et al. 2025). Barrett et al. (2025) subsequently showed that
for all graphs on 8 vertices
with the exception of a set of 7 graphs
denoted 
to 
and which are illustrated above. As it turns out, 
is the graph complement
of the 3-Plummer-Toft graph.
for every graph 
and 
. They also establish the
corresponding monotonicity for the b-forcing
number, whose rule allows a colored vertex with
at most 
uncolored neighbors to force all of those neighbors
at once, and for the positive
semidefinite zero forcing number.
for 
and 
for 
, where 
and 
is the maximum nullity
(Abiad et al. 2026). The family formulas are drawn from earlier results on
minimum rank.
with 
and 
, the general upper bound
is 
(Alameda et al. 2018, p. 58). Previously known exact values include the
prism graph values 
and 
for 
(Alameda et al. 2018, p. 58). Alameda et
al. (2018, p. 60) also proved 
for every integer 
. Rashidi et al. (2020, pp. 185 and
189-190) proved 
for 
and 
for 
. Krishnan et al. (2026) give the exact
values for all members of the generalized
Petersen graph subfamilies 
and 
: 
for every 
and 
for every 
. The smaller values of
for 
, 8, ..., 12 are 6, 6, 6, 8, 7, 7, and those of 
for 
, 10, ..., 17 are 6, 6, 7, 6, 8, 8, 9, 8, 9. Krishnan
et al. (2026) report AI assistance with computational code and Lean development
for the 
stabilization theorem.
and 
."
16 Oct 2026.