Minimum Forcing Set in Graph Theory

• Minimum forcing set is defined as the smallest subset of vertices from which the entire graph can be turned black by applying the color-change rule.
• The topic encompasses iterative dynamics, structural bounds, and exact values across various graph families, with key metrics like the zero forcing number and iteration index.
• Applications extend to matrix theory, network controllability, and even non-graph settings such as origami and matching problems, highlighting its broad impact.

In graph theory, a minimum forcing set usually denotes a zero forcing set of minimum cardinality: an initial set of black vertices from which the entire graph is forced black by repeated applications of the color-change rule. For a graph $G$, this minimum size is the zero forcing number $Z(G)$, and a zero forcing set of size $Z(G)$ is often called a $Z(G)$-set or minimum zero forcing set (Chilakamarri et al., 2011). Later work emphasized that “minimum” is distinct from “minimal”: every minimum zero forcing set is inclusion-minimal, but a minimal zero forcing set can be strictly larger than $Z(G)$ (Brimkov et al., 2022).

1. Classical zero forcing framework

Let $G=(V(G),E(G))$ be a graph whose vertices are colored black or white. In the standard color-change rule, a black vertex $u$ forces a white vertex $w$ if $w$ is the only white neighbor of $u$; this is written $Z(G)$0 (Chilakamarri et al., 2011). A subset $Z(G)$1 is a zero forcing set if, starting with $Z(G)$2 black and all other vertices white, repeated global applications of this rule eventually turn every vertex black. The zero forcing number is

$Z(G)$3

and a zero forcing set of cardinality $Z(G)$4 is a minimum forcing set in the standard zero forcing sense (Chilakamarri et al., 2011).

The process admits a discrete dynamical formulation. For $Z(G)$5, the functions $Z(G)$6 are defined recursively by

$Z(G)$7

and

$Z(G)$8

Here $Z(G)$9 is the open neighborhood and $Z(G)$0 the closed neighborhood (Chilakamarri et al., 2011).

Associated notions include the $Z(G)$1-th derived set

$Z(G)$2

the nested black-vertex sets $Z(G)$3, chronological lists of forces, forcing chains, maximal forcing chains, and reversals. If a chronological list of forces is fixed, the reversal of $Z(G)$4 is the set of last vertices of maximal forcing chains; the reversal of a zero forcing set is also a zero forcing set (Chilakamarri et al., 2011).

Minimality and minimum size are different conditions. A minimal zero forcing set is inclusion-minimal, whereas a minimum zero forcing set has size exactly $Z(G)$5 (Brimkov et al., 2022). A graph is called well-forced if every minimal zero forcing set has cardinality $Z(G)$6 (Grood et al., 2023).

2. Structural properties and matrix-theoretic context

Several basic inequalities organize the theory. For any graph $Z(G)$7, the minimum degree satisfies $Z(G)$8, and the path cover number satisfies $Z(G)$9 (Chilakamarri et al., 2011). For trees, these parameters coincide: if $Z(G)$0 is a tree, then $Z(G)$1 (Chilakamarri et al., 2011). Equivalently, for trees one also has $Z(G)$2, where $Z(G)$3 is the minimum number of vertex-disjoint induced paths covering the vertex set (Grood et al., 2023).

Zero forcing sets are generally non-unique. The reversal property shows that different chronological lists can generate different zero forcing sets, and if a graph has a unique zero forcing set, then it must be edgeless (Chilakamarri et al., 2011). The parameter is also stable under small deletions: for a vertex $Z(G)$4,

$Z(G)$5

and for an edge $Z(G)$6,

$Z(G)$7

(Chilakamarri et al., 2011).

The principal algebraic motivation comes from minimum rank and maximum nullity. If $Z(G)$8 is a symmetric matrix with graph $Z(G)$9 and $Z(G)$0 vanishes on a zero forcing set $Z(G)$1, then $Z(G)$2 must vanish everywhere; consequently zero forcing gives an upper bound on maximum nullity and hence relates directly to minimum rank (Chilakamarri et al., 2011). In the formulation of the random-graph study, if $Z(G)$3 denotes maximum nullity over symmetric real matrices described by $Z(G)$4, then $Z(G)$5 for all graphs (Kalinowski et al., 2017).

On trees, the correspondence with path covers is especially rigid. Given a minimum path cover, choosing one end-vertex from each path yields a minimum zero forcing set, and given a minimum zero forcing set, its maximal forcing chains form a minimum path cover (Grood et al., 2023). This is one reason trees serve as the main testing ground for finer questions such as irrelevance of vertices, well-forcedness, and the gap between minimal and minimum zero forcing sets.

3. Iteration index and exact values for standard families

The 2011 introduction of the iteration index refined the size-only viewpoint by measuring the number of global forcing rounds used by a minimum forcing set. For a zero forcing set $Z(G)$6,

$Z(G)$7

and the iteration index of the graph is

$Z(G)$8

Thus $Z(G)$9 counts the number of global steps needed to blacken all vertices from $G=(V(G),E(G))$0, and $G=(V(G),E(G))$1 chooses the smallest such value among minimum forcing sets (Chilakamarri et al., 2011).

For graphs without isolated vertices, the iteration index satisfies the fundamental bounds

$G=(V(G),E(G))$2

Both bounds are sharp. The lower bound is attained by $G=(V(G),E(G))$3, where $G=(V(G),E(G))$4, and by grid graphs $G=(V(G),E(G))$5 with $G=(V(G),E(G))$6, where $G=(V(G),E(G))$7. The upper bound is attained by the graph obtained by joining the center of a star $G=(V(G),E(G))$8 to an end-vertex of a path $G=(V(G),E(G))$9, for which $u$0 (Chilakamarri et al., 2011).

The value of $u$1 can depend on which minimum forcing set is chosen. In $u$2, one minimum forcing set has iteration index $u$3, while another has iteration index $u$4; this is precisely why $u$5 is defined by minimizing over all $u$6-sets (Chilakamarri et al., 2011).

The standard exact values are as follows.

Graph family	$u$7	$u$8
$u$9 $w$0	$w$1	$w$2
$w$3 $w$4	$w$5	$w$6
$w$7 $w$8	$w$9	$w$0
$w$1 $w$2	$w$3	$w$4
$w$5 $w$6	$w$7	$w$8

These values come with explicit constructions. For $w$9, only an end-vertex is a minimum forcing set, and forcing proceeds as a single chain of length $u$0. For $u$1, a minimum forcing set is an adjacent pair, and forcing propagates in both directions around the cycle. For $u$2, all but one vertex black forces the remaining vertex in one round. For $u$3, taking all leaves except one gives a two-round process: a leaf forces the center, and then the center forces the last leaf (Chilakamarri et al., 2011).

For Cartesian products, the paper gives several exact formulas. If $u$4, then

$u$5

If $u$6, then

$u$7

For $u$8 with $u$9 and $Z(G)$00,

$Z(G)$01

and

$Z(G)$02

For $Z(G)$03 with $Z(G)$04 and $Z(G)$05,

$Z(G)$06

For $Z(G)$07 with $Z(G)$08,

$Z(G)$09

(Chilakamarri et al., 2011).

The same work also gives upper bounds for triangular grids and king grids and an exact formula for bouquets of circles. If $Z(G)$10 is a bouquet of $Z(G)$11 cycles with $Z(G)$12 and $Z(G)$13, then

$Z(G)$14

(Chilakamarri et al., 2011).

4. Extremal, probabilistic, and spectral estimates

Beyond exact families, the zero forcing number has been studied on large structural classes. One line of work treats graphs of large girth, $Z(G)$15-free graphs for a fixed bipartite graph $Z(G)$16, random graphs, and pseudorandom graphs (Kalinowski et al., 2017).

For binomial random graphs $Z(G)$17 with

$Z(G)$18

one has with high probability

$Z(G)$19

In particular, for $Z(G)$20,

$Z(G)$21

and for $Z(G)$22,

$Z(G)$23

(Kalinowski et al., 2017).

For $Z(G)$24-graphs, the spectral estimates are of the form

$Z(G)$25

and

$Z(G)$26

where $Z(G)$27 is the smallest eigenvalue (Kalinowski et al., 2017). These bounds show that in good expanders the zero forcing number remains close to the order of the graph.

The random-graph results also sharpen the distinction between zero forcing and matrix-theoretic nullity in typical graphs. In contrast to the asymptotic formula above for $Z(G)$28, the same study notes that $Z(G)$29 lies in $Z(G)$30 with high probability, so for most graphs $Z(G)$31 and $Z(G)$32 are far apart (Kalinowski et al., 2017). This suggests that minimum forcing sets are often much larger than nullity-based obstructions alone would indicate.

5. Variants of minimum forcing sets in graphs

A substantial part of the literature replaces the standard color-change rule or adds constraints on the initial set. The most developed variants are total forcing, connected forcing, skew forcing, positive semidefinite forcing, edge forcing, and dom-forcing.

A total forcing set is a zero forcing set whose induced subgraph has no isolated vertices; its minimum size is $Z(G)$33 (Davila et al., 2017). For connected graphs of order $Z(G)$34 and maximum degree $Z(G)$35,

$Z(G)$36

with equality if and only if $Z(G)$37 or $Z(G)$38 (Davila et al., 2017). On trees, the parameter is closely tied to path covers and matchings: for every nontrivial tree $Z(G)$39,

$Z(G)$40

and the extremal trees for both bounds are characterized (Davila et al., 2018). A complementary tree result states that if $Z(G)$41 is a non-trivial tree with $Z(G)$42 leaves, then $Z(G)$43, and consequently $Z(G)$44 (Davila et al., 2017). In connected claw-free cubic graphs of order $Z(G)$45, one has

$Z(G)$46

with equality exactly for the prism $Z(G)$47 and the diamond-necklace family (Davila et al., 2017).

A connected forcing set is a zero forcing set inducing a connected subgraph; its minimum size is $Z(G)$48 (Davila et al., 2016). For connected graphs,

$Z(G)$49

and exact values include $Z(G)$50, $Z(G)$51, and $Z(G)$52 (Davila et al., 2016). The connected variant is computationally harder than it may first appear: the decision problem for connected zero forcing is NP-complete (Brimkov, 2016). It also satisfies sharper lower bounds in terms of girth and minimum degree, such as

$Z(G)$53

when $Z(G)$54 and $Z(G)$55 (Davila et al., 2016).

Skew zero forcing relaxes the rule so that any vertex, black or white, can force when it has exactly one white neighbor. The minimum size is the skew zero forcing number $Z(G)$56 (DeAlba, 2014). A connected graph $Z(G)$57 is complete multipartite if and only if

$Z(G)$58

and in many bipartite families minimum skew zero forcing sets are realized as the unsaturated sets of maximum matchings (DeAlba, 2014). In bipartite graphs in which all maximum matchings are uniquely restricted, the minimum skew zero forcing sets are precisely the bases of the dual of the matching matroid (DeAlba, 2014).

The positive semidefinite and skew variants also have a reconfiguration theory. For a tree $Z(G)$59 on $Z(G)$60 vertices, the reconfiguration graph of minimum PSD forcing sets under token exchange is $Z(G)$61, and under token sliding it is $Z(G)$62 itself (Bong et al., 7 Jan 2025). For skew forcing on trees, the token-exchange reconfiguration graph is connected, whereas the token-sliding reconfiguration graph has no edges (Bong et al., 7 Jan 2025). These results make explicit how “minimum forcing set” can be studied as a configuration space, not only as a static optimum.

Edge forcing begins from a matching of initially active edges rather than a set of vertices. If $Z(G)$63 is a set of independent edges and the endpoint set $Z(G)$64 zero forces all of $Z(G)$65, then $Z(G)$66 is an edge-forcing set; its minimum cardinality is $Z(G)$67 (G. et al., 2021). The decision problem is NP-complete. For butterfly networks, $Z(G)$68 has no edge-forcing set, while

$Z(G)$69

(G. et al., 2021).

A more recent hybrid notion is the connected dom-forcing set, which must be simultaneously a connected dominating set and a connected zero forcing set; its minimum size is $Z(G)$70 (P et al., 16 May 2025). The exact values include

$Z(G)$71

for the graph classes treated in that paper (P et al., 16 May 2025).

6. Extensions beyond vertex zero forcing

The expression “minimum forcing set” also appears in several non-zero-forcing settings, where the common theme is uniqueness enforced by prescribing a smallest subset.

In mathematical origami, a forcing set is a subset of creases whose mountain-valley assignments determine the entire locally flat-foldable pattern. For an $Z(G)$72 Miura-ori, locally flat-foldable assignments correspond bijectively to $Z(G)$73-colorings of a grid graph, and minimum forcing sets correspond to minimum feedback arc sets in an associated planar digraph (Ballinger et al., 2014). The size satisfies

$Z(G)$74

and for the standard Miura-ori assignment the upper bound is attained: $Z(G)$75 (Ballinger et al., 2014). For one-dimensional origami, there is a linear-time algorithm, and the minimum forcing set size is exactly $Z(G)$76, where $Z(G)$77 is the number of monocrimps in an exhaustive crimp sequence and $Z(G)$78 is the number of end creases (Damian et al., 2017).

For perfect matchings, a forcing set of a perfect matching $Z(G)$79 is a subset $Z(G)$80 such that $Z(G)$81 is the only perfect matching containing $Z(G)$82; the minimum size is the forcing number $Z(G)$83 (Diwan, 2017). In the hypercube $Z(G)$84, every perfect matching satisfies

$Z(G)$85

which resolved a conjecture of Pachter and Kim for all $Z(G)$86 (Diwan, 2017).

For maximal matchings, the relevant notion is global forcing: a set $Z(G)$87 is a global forcing set if distinct maximal matchings have distinct intersections with $Z(G)$88 (Klavžar et al., 2021). In corona products, the paper establishes lower and upper bounds and gives an integer linear programming formulation for computing the minimum global forcing set (Klavžar et al., 2021).

For optimization problems such as minimum spanning trees and shortest paths, a forcing set is a set of elements that lies in exactly one optimum solution (Gima et al., 29 Sep 2025). In that setting, the minimum forcing set and minimum anti-forcing set for minimum spanning trees are both solvable in polynomial time; the minimum forcing set for shortest $Z(G)$89-$Z(G)$90 paths is also polynomial-time solvable, whereas the minimum anti-forcing set for shortest $Z(G)$91-$Z(G)$92 paths is NP-hard (Gima et al., 29 Sep 2025). This suggests that the forcing-set paradigm extends beyond graph coloring dynamics to broader uniqueness problems in combinatorial optimization.

Across these domains, the phrase “minimum forcing set” retains a consistent core meaning: the smallest prescribed subset that eliminates all but one admissible completion. In zero forcing, that completion is a global propagation process; in origami, it is a flat-foldable mountain-valley assignment; in matching and optimization settings, it is a unique optimum or unique feasible structure.