Minimum Anti-Forcing Set in Graphs

• Minimum anti-forcing set is the smallest set of non-matching edges whose removal guarantees a unique perfect matching by hitting every alternating cycle.
• Researchers compute these sets using cycle transversal methods, planar duality, and feedback arc set transformations in both planar and non-planar contexts.
• The concept finds applications in chemical graph theory, resonance structure analysis of benzenoid systems, and extends to optimization in spanning trees and matroid bases.

A minimum anti-forcing set is, in its original graph-theoretic sense, a minimum-cardinality set of edges excluded from a fixed perfect matching so that the matching becomes unique after those edges are removed. Formally, if $G$ has a perfect matching $M$, then a set $S \subseteq E(G)\setminus M$ is an anti-forcing set of $M$ when $G-S$ has $M$ as its unique perfect matching, and any such set with $|S|=af(G,M)$ is a minimum anti-forcing set. The notion sits at the intersection of matching theory, cycle transversals, planar duality, and chemical graph theory, and later admits a broader formulation for general optimal-solution families in combinatorial optimization (Lei et al., 2014, Gima et al., 29 Sep 2025).

1. Definitions, variants, and the cycle-hitting viewpoint

Let $G$ be a finite simple graph with at least one perfect matching. For a perfect matching $M$, the anti-forcing number is

$$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$

A “minimum anti-forcing set” is any $M$0 attaining this minimum. At graph level, the literature distinguishes the minimum and maximum over perfect matchings: one convention writes

$M$1

This distinction is essential: $M$2 is attached to a fixed matching, whereas $M$3 and $M$4 summarize the full matching space (Lei et al., 2014).

The basic structural object is an $M$5-alternating cycle, namely a cycle whose edges alternate between $M$6 and $M$7. The fundamental characterization is that $M$8 is an anti-forcing set of $M$9 if and only if $S \subseteq E(G)\setminus M$0 contains at least one edge of every $S \subseteq E(G)\setminus M$1-alternating cycle. Equivalently, $S \subseteq E(G)\setminus M$2 is the minimum size of a hitting set for the hypergraph of $S \subseteq E(G)\setminus M$3-alternating cycles. This makes minimum anti-forcing a transversal problem rather than a direct uniqueness problem (Lei et al., 2014).

Related forcing notions use inclusion rather than exclusion. For a fixed perfect matching, a forcing set is a subset of $S \subseteq E(G)\setminus M$4 meeting every $S \subseteq E(G)\setminus M$5-alternating cycle, while a global forcing set is an edge set distinguishing all perfect matchings. In the global theory, “nice cycles” also appear: a cycle $S \subseteq E(G)\setminus M$6 is nice if $S \subseteq E(G)\setminus M$7 has a perfect matching, equivalently if two perfect matchings differ exactly on $S \subseteq E(G)\setminus M$8 (Zhang et al., 2020).

The same exclusion-based schema extends beyond perfect matchings. If $S \subseteq E(G)\setminus M$9 is the set of optimal solutions of a combinatorial optimization problem and $M$0, then $M$1 is an anti-forcing set for $M$2 if $M$3 and every other $M$4 intersects $M$5. Equivalently, $M$6 is the unique optimal solution disjoint from $M$7 (Gima et al., 29 Sep 2025).

2. Planar bipartite graphs and the minimax theorem

For plane bipartite graphs, anti-forcing admits a sharp packing-transversal formulation. Two $M$8-alternating cycles are called compatible if they are disjoint or intersect only at edges of $M$9. Let $G-S$0 denote the maximum size of a compatible $G-S$1-alternating set. Then, for a planar bipartite graph with perfect matching $G-S$2,

$G-S$3

Thus, in this setting, a minimum anti-forcing set has the same cardinality as a maximum compatible packing of alternating cycles (Lei et al., 2014).

The proof uses an orientation-contraction transformation. Edges in $G-S$4 are oriented from white to black, edges outside $G-S$5 from black to white, and then all $G-S$6-edges are contracted. In the resulting planar digraph, $G-S$7-alternating cycles correspond to directed cycles, and anti-forcing sets correspond to feedback arc sets. The Lucchesi–Younger theorem then identifies the minimum feedback arc set size with the maximum number of arc-disjoint directed cycles, which translates back to the equality $G-S$8 (Lei et al., 2014).

This minimax theorem gives a constructive interpretation of minimum anti-forcing sets. In plane bipartite graphs, the problem can be approached by computing either a maximum compatible family of $G-S$9-alternating cycles or a minimum feedback arc set in the contracted planar digraph. The result is exact, but the cited work does not state an explicit complexity bound for the general plane-bipartite computation; its algorithmic content is primarily structural and existential (Lei et al., 2014).

The planar bipartite equality is not universal. Outside the plane bipartite class, the relation $M$0 may fail. The dodecahedron is cited as a counterexample: for a suitable perfect matching, there are at most three compatible $M$1-alternating cycles, but the anti-forcing number is at least four. A common misconception is therefore to treat compatible alternating-cycle packing as a general formula; it is a theorem of the planar bipartite regime, not of arbitrary graphs (Lei et al., 2014).

3. Relations with forcing and global forcing invariants

The comparison between minimum anti-forcing sets and forcing-type invariants is most explicit through the global forcing number

$M$2

A set is global forcing exactly when it intersects each nice cycle of $M$3. The paper on global forcing versus maximum anti-forcing establishes a bridge lemma: if $M$4 is a minimum global forcing set, then there exists $M$5 such that $M$6 has a unique perfect matching. This connects minimum global cycle transversals to anti-forcing-type uniqueness certificates (Zhang et al., 2020).

For bipartite graphs with a perfect matching,

$M$7

Since $M$8, it follows that

$M$9

in the bipartite case. The same inequality extends to the class $|S|=af(G,M)$0 of graphs with a perfect matching and no two disjoint odd cycles $|S|=af(G,M)$1 such that $|S|=af(G,M)$2 still has a perfect matching. It also holds for graphs whose perfect matching polytopes consist of non-negative 1-regular vectors; for bricks, this is equivalent to solidity (Zhang et al., 2020).

The quantitative gap $|S|=af(G,M)$3 is sharply bounded. For a connected bipartite graph with $|S|=af(G,M)$4 vertices,

$|S|=af(G,M)$5

with equality on the right if and only if $|S|=af(G,M)$6. For connected graphs with $|S|=af(G,M)$7 vertices, not necessarily bipartite,

$|S|=af(G,M)$8

the upper bound being attained by $|S|=af(G,M)$9, while the lower bound is tight only for $G$0 (Zhang et al., 2020).

These bounds clarify the status of minimum anti-forcing sets. In bipartite and $G$1-graphs, minimum anti-forcing sets for any fixed perfect matching are never larger than minimum global forcing sets. In general non-bipartite graphs, however, the direction can reverse. The matching covered family $G$2 satisfies

$G$3

showing that anti-forcing can exceed global forcing by an arbitrarily large additive amount outside the controlled classes (Zhang et al., 2020).

4. Hexagonal systems, benzenoid chemistry, and fullerene classes

In chemical graph theory, perfect matchings are Kekulé structures, and anti-forcing measures how many non-Kekulé edges must be deleted to leave a unique Kekulé structure. For hexagonal systems $G$4, the maximum anti-forcing number equals the Fries number: $G$5 Combined with the known identity $G$6 for the maximum forcing number, this yields the Fries–Clar inequalities

$G$7

Accordingly, maximum anti-forcing sets in benzenoid graphs are tied to classical resonance indices rather than being merely auxiliary matching invariants (Lei et al., 2014).

The cata-condensed subclass admits a more refined description. If $G$8 is a cata-condensed hexagonal system, then the anti-forcing spectrum

$G$9

is continuous: it is an integer interval

$M$0

Here the inner dual $M$1 is a tree, all cycles are nice, and one can choose a maximum non-crossing compatible $M$2-alternating set with minimal $M$3-index; such a set contains all $M$4-alternating hexagons. For these systems, a minimum anti-forcing set for a fixed $M$5 can be constructed by selecting one non-$M$6 edge from each cycle in a maximum non-crossing compatible $M$7-alternating set (Deng et al., 2014).

Several explicit values are known. For a single hexagon, $M$8. For a linear chain with $M$9 hexagons,

$$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$0

Hence benzene has anti-forcing spectrum $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$1, while naphthalene, anthracene, and all longer linear polyacenes have spectrum $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$2. The strict inequality

$$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$3

holds for cata-condensed systems with at least two hexagons (Deng et al., 2014).

Fullerenes exhibit a different anti-forcing profile. For fullerene graphs $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$4, the cited work reports the lower bound

$$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$5

and discusses the families with $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$6. It also states that, except for the exceptional fullerene $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$7, every fullerene with anti-forcing number $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$8 has minimum forcing number $$af(G,M)=\min\{\,|S|:S\subseteq E(G)\setminus M,\ G-S\text{ has a unique perfect matching}\,\}.$$9; nanotube fullerenes of type $M$00 are examples. The same source notes that $M$01 satisfies $M$02 while being exceptional for the forcing-number bound (Shi et al., 2018).

5. Extremal values, spectra, and structural decompositions

For connected graphs with a perfect matching, the maximum anti-forcing number is bounded by the cyclomatic number

$M$03

If $M$04 is non-bipartite, then the inequality is strict: $M$05 The proof strategy deletes non-$M$06 cycle edges until a spanning tree remains; the deleted edges form an anti-forcing set, and odd cycles explain the strictness in the non-bipartite case (Deng et al., 2016).

The extremal equality $M$07 is structurally rigid. It holds precisely for planar bipartite graphs whose blocks are either fixed edges or normal components admitting a suitable bipartite ear decomposition. In the elementary case with $M$08, there is a unique perfect matching $M$09 for which $M$10. This identifies extremal anti-forcing with a compatible fundamental cycle basis of the cycle space (Deng et al., 2016).

At the opposite end, plane elementary bipartite graphs with minimum anti-forcing number one are characterized by a local face configuration. Such a graph has an anti-forcing edge if and only if there exists a perfect matching $M$11 such that the graph has exactly two $M$12-resonant faces whose boundaries have a common path of length at least $M$13; any non-$M$14 edge on that common path is an anti-forcing edge. Thus $M$15 is governed by a very specific resonant-face overlap pattern rather than by global sparsity alone (Deng et al., 2016).

Even polygonal chains provide a particularly explicit spectral theory. Their anti-forcing spectrum is an integer interval, and both endpoints are computable in linear time. If $M$16 is the segment decomposition, then

$M$17

If $M$18 is the all-kink decomposition, then

$M$19

Consequences include the formulas $M$20 for all-kink chains without 4-cycles and $M$21 for straight chains of $M$22 squares. One worked example in the cited paper yields

$M$23

These results show that minimum anti-forcing sets can sometimes be located by deterministic decomposition rules rather than by unrestricted cycle hitting (Deng et al., 2016).

6. Generalization to shortest paths, spanning trees, and matroid bases

Later work extends anti-forcing from perfect matchings to arbitrary optimal-solution families. If $M$24 is the set of optimal solutions of a weighted combinatorial problem, a set $M$25 is anti-forcing for $M$26 when $M$27 and every other optimal solution intersects $M$28. In this form, minimum anti-forcing becomes a general tie-breaking problem by exclusion rather than a matching-specific invariant (Gima et al., 29 Sep 2025).

For shortest $M$29-$M$30 paths, the complexity landscape is asymmetric. Minimum forcing is polynomial, solvable in $M$31 time by reducing to the shortest-path DAG and dynamic programming. By contrast, minimum anti-forcing for shortest paths is NP-complete even on undirected unweighted graphs. The cited reduction is from Vertex Cover. Two tractable special cases are also identified: when a specific shortest path $M$32 is given, a minimum anti-forcing set disjoint from $M$33 can be found in polynomial time via minimum multiway cut on the DAG $M$34; and on bounded-treewidth graphs, the problem is solvable in linear time through MSO$M$35 formulations and Courcelle’s theorem (Gima et al., 29 Sep 2025).

For minimum spanning trees, both forcing and anti-forcing are polynomial-time computable. The paper gives $M$36-time Kruskal-style algorithms operating on equal-weight tie classes. For anti-forcing, within each tie class $M$37, one computes a maximal forest $M$38, adds $M$39 except self-loops to the anti-forcing set, contracts $M$40, and iterates. The same perspective extends to minimum-weight bases of a matroid presented by an independence oracle, where minimum forcing and anti-forcing sets are computable in polynomial time, and forcing in a matroid is dual to anti-forcing for maximum bases in the dual matroid (Gima et al., 29 Sep 2025).

This broader optimization framework suggests a conceptual continuity with the original matching theory. In all cases, a minimum anti-forcing set is an exclusion set of minimum cardinality that destroys all alternative optimal solutions while leaving one designated optimum intact. What varies is the structure of the obstruction family: alternating cycles for perfect matchings, recombinable subpaths in shortest-path DAGs, and exchangeable equal-weight choices in spanning-tree and matroid tie classes (Gima et al., 29 Sep 2025).