Min-Forced Vertices in Graphs

• Min-forced vertices are a family of invariants that identify elements appearing in every minimum solution across various graph optimization problems.
• They are applied in settings like locating-dominating codes, metric bases, zero forcing, and perfect matchings to certify uniqueness and guide propagation processes.
• Studying these vertices reveals structural properties and computational challenges, with implications in areas from efficient algorithms to chemical graph theory and origami.

“Min-forced vertices” is not a single invariant but a family of closely related notions concerning elements that are either unavoidable in every minimum solution or are counted by a minimum forcing process. In one line of work, the term refers explicitly to vertices that belong to every minimum locating-dominating code (Junnila et al., 1 Sep 2025). In another, the corresponding notion appears under different names, such as basis forced vertices for metric bases (Hakanen et al., 2021). In zero forcing and total forcing on graphs, the same viewpoint becomes a minimization problem over initially colored vertex sets (Davila et al., 2017). In perfect matching theory and related optimization problems, the analogous objects are usually edges rather than vertices: a forcing set is a minimum subset that uniquely determines a perfect matching, a shortest path, or a minimum spanning tree (Liu et al., 2020, Gima et al., 29 Sep 2025). The subject therefore spans propagation processes, uniqueness certificates, extremal graph structure, and computational complexity.

1. Terminological scope and core formulations

The common backbone is a forcing principle: a small partial specification determines a unique global object. In the general formulation used for shortest paths and minimum spanning trees, if $E$ is a ground set and $\Pi \subseteq 2^E$ is a family of feasible solutions, then $S \subseteq E$ is a forcing set for $X \in \Pi$ when $X$ is the unique solution containing $S$; dually, $S$ is anti-forcing when $X$ is the unique solution disjoint from $S$ (Gima et al., 29 Sep 2025). This definition subsumes several older graph-theoretic forcing notions.

A second, more vertex-centric usage concerns unavoidable vertices in minimum structures. A basis forced vertex is a vertex contained in every metric basis of a graph (Hakanen et al., 2021). A min-forced vertex, in the locating-dominating setting, is a vertex that belongs to every minimum locating-dominating code (Junnila et al., 1 Sep 2025). These are not propagation parameters but “core” vertices of all optimum solutions.

A third usage, explicitly identified with the “min-forced vertices” viewpoint in trees, treats forcing as a dynamic coloring process. A forcing set is an initial colored vertex set from which repeated applications of the rule “a colored vertex with exactly one non-colored neighbor forces that neighbor” eventually color the whole graph; a total forcing set adds the condition that the induced subgraph on the initial set has no isolated vertices (Davila et al., 2017). Here the problem is not whether a specific vertex is unavoidable, but how few initial vertices suffice.

Setting	Object being forced	Minimum notion
Zero forcing / total forcing	Initially colored vertices	$F(G)$, $\Pi \subseteq 2^E$0 (Davila et al., 2017)
Perfect matchings	Edges of a matching	$\Pi \subseteq 2^E$1, $\Pi \subseteq 2^E$2, $\Pi \subseteq 2^E$3 (Liu et al., 2020)
Metric dimension	Vertices in resolving sets	Basis forced vertices (Hakanen et al., 2021)
Locating-dominating codes	Vertices in LD-codes	Min-forced vertices (Junnila et al., 1 Sep 2025)
Shortest paths / MSTs	Edges in optimal solutions	Minimum forcing / anti-forcing sets (Gima et al., 29 Sep 2025)
1D origami	Creases with fixed MV labels	Minimum forcing sets (Damian et al., 2017)

This variety matters conceptually. Some papers study the size of a minimum forcing set; others study which vertices belong to all minimum sets. The shared theme is uniqueness under partial information, but the formal invariants differ.

2. Propagation-based forcing on vertices: zero forcing and total forcing

For trees, the most direct “minimum initially forced vertices” formulation is total forcing. A dynamic coloring starts from an initial set $\Pi \subseteq 2^E$4, and whenever a colored vertex has exactly one non-colored neighbor, it forces that neighbor to become colored. The forcing number $\Pi \subseteq 2^E$5 is the minimum size of a forcing set. The total forcing number $\Pi \subseteq 2^E$6 is the minimum size of a forcing set whose induced subgraph has no isolated vertices (Davila et al., 2017).

The tree case is sharply structured. If $\Pi \subseteq 2^E$7 is a tree of order $\Pi \subseteq 2^E$8 with maximum degree $\Pi \subseteq 2^E$9, then

$S \subseteq E$0

with equality exactly for a family $S \subseteq E$1 built from star-like underlying subtrees whose centers are independent and are strong support vertices (Davila et al., 2017). If $S \subseteq E$2 is the number of leaves of a non-trivial tree, then

$S \subseteq E$3

with equality exactly for a recursively defined family $S \subseteq E$4 obtained from $S \subseteq E$5 by five operations $S \subseteq E$6 (Davila et al., 2017).

A decisive structural consequence is that total forcing is strictly more demanding than ordinary forcing on non-trivial trees: $S \subseteq E$7 with equality exactly for trees in a family $S \subseteq E$8, namely paths and trees whose trimmed form is a star (Davila et al., 2017). The gap can be arbitrarily large: for every integer $S \subseteq E$9, there exists a tree $X \in \Pi$0 with

$X \in \Pi$1

The main mechanisms behind these bounds are trimming invariance, subdivision invariance, and the behavior of strong support vertices. Every total forcing set in an isolate-free graph contains every strong support vertex and all except possibly one leaf neighbor of each strong support vertex (Davila et al., 2017). In this sense, leaves and support structure act as local forcing bottlenecks. For the min-forced-vertices interpretation, this means that the smallest admissible initial set is controlled simultaneously by leaf count, maximum degree, and the requirement that the initial set itself be isolate-free.

3. Edge-based forcing in perfect matching theory

In perfect matching theory, the basic object is a forcing set of edges. For a graph $X \in \Pi$2 with a perfect matching $X \in \Pi$3, a subset $X \in \Pi$4 is forcing if no other perfect matching of $X \in \Pi$5 contains all edges of $X \in \Pi$6. The forcing number $X \in \Pi$7 is the minimum size of such a subset; the minimum and maximum forcing numbers of the graph are

$X \in \Pi$8

For a graph of order $X \in \Pi$9 with a perfect matching, one always has

$X$0

(Liu et al., 2020, Liu et al., 2021).

Che and Chen asked for a characterization of graphs with $X$1. This was completely solved by showing that $X$2 holds exactly for complete multipartite graphs whose partite sets all have size at most $X$3, and for graphs obtained from $X$4 by adding arbitrary edges within the same partite set (Liu et al., 2020). In the bipartite case, the earlier Che–Chen characterization is recovered: $X$5 if and only if $X$6 (Liu et al., 2020, Liu et al., 2021).

The extremal $X$7 phenomenon is driven by alternating cycles. A key criterion states that a perfect matching $X$8 satisfies $X$9 if and only if for any two distinct edges $S$0, the induced subgraph on their four endpoints contains an $S$1-alternating cycle (Liu et al., 2020). This is the rigid end of the spectrum: every pair of matching edges lies inside a local alternating configuration, so one must fix almost all of the matching to eliminate ambiguity.

Subsequent work sharpened the extremal picture. For graphs of order $S$2 with $S$3, one has

$S$4

with equality exactly for graphs of the form $S$5; for bipartite graphs,

$S$6

with equality exactly for $S$7 (Liu et al., 2021). These formulas generalize the classical unique-perfect-matching bounds of Hetyei and Lovász–Plummer from $S$8 to arbitrary $S$9.

At the opposite end, Liu and Zhang’s conjecture on maximum forcing number was confirmed: $S$0 (Liu et al., 28 Dec 2025). In bipartite graphs with $S$1, the possible minimum forcing numbers form an interval: $S$2 (Liu et al., 28 Dec 2025). The forcing spectrum is therefore both rigid and surprisingly continuous in extremal regimes.

A specialized chemical application appears in fullerene graphs. Except for a unique fullerene $S$3 with minimum forcing number $S$4, every fullerene has minimum forcing number at least $S$5, and all fullerenes with minimum forcing number $S$6 are characterized by a patch-based construction using generalized patches, distance-arrays, and operations $S$7 (Shi et al., 2018). In particular, nanotube fullerenes of type $S$8 satisfy $S$9.

4. Forcing beyond matchings: origami, shortest paths, and minimum spanning trees

The forcing paradigm extends well beyond graph matchings. In one-dimensional origami, a forcing set is a subset of creases whose mountain-valley labels determine the entire foldable assignment. For a fixed foldable one-dimensional MV pattern, a linear-time algorithm constructs a minimum forcing set via a crimp forest that encodes all crimpable sequences and their dependencies (Damian et al., 2017). The minimum size equals $X$0, where $X$1 is the number of monocrimps in an exhaustive crimp sequence and $X$2 is the number of end creases (Damian et al., 2017).

For shortest $X$3-$X$4 paths and minimum spanning trees, forcing and anti-forcing were studied in a general optimization framework. For shortest paths, minimum forcing is tractable: the decision problem can be solved in time $X$5 after reduction to the DAG of shortest-path edges (Gima et al., 29 Sep 2025). Minimum anti-forcing for shortest paths is NP-complete even for undirected unweighted graphs, although it is polynomial-time solvable for a fixed shortest path and for bounded-treewidth graphs (Gima et al., 29 Sep 2025). For minimum spanning trees, both forcing and anti-forcing are polynomial-time solvable; in fact the paper gives $X$6 algorithms via a matroidal viewpoint and duality between forcing and anti-forcing (Gima et al., 29 Sep 2025).

These results show that forcing complexity is highly problem-dependent. For perfect matchings, minimum forcing is NP-hard in general (Gima et al., 29 Sep 2025); for shortest paths, forcing is easy but anti-forcing is hard (Gima et al., 29 Sep 2025); for minimum spanning trees, both are tractable (Gima et al., 29 Sep 2025). A plausible implication is that “min-forced” phenomena depend less on solution size alone than on the exchange structure of the underlying feasible family.

5. Vertices contained in every minimum solution

The most literal vertex-based form of the topic appears in locating-dominating codes. A min-forced vertex is a vertex that belongs to every minimum locating-dominating code (Junnila et al., 1 Sep 2025). If $X$7 is a connected nontrivial graph of order $X$8, and $X$9 denotes the number of min-forced vertices, then

$S$0

which implies

$S$1

Both bounds are attained (Junnila et al., 1 Sep 2025). The proof uses a color graph attached to a minimum locating-dominating code and shows that each min-forced codeword must contribute at least two edges of its color in the non-codeword part of that graph; cactus bounds then limit how many such colors can coexist (Junnila et al., 1 Sep 2025).

The same paper gives a local characterization: a vertex $S$2 is min-forced if and only if $S$3 is isolated, or $S$4, or $S$5 and no minimum locating-dominating code of $S$6 both dominates and uniquely identifies $S$7 when viewed back in $S$8 (Junnila et al., 1 Sep 2025). It also proves that deciding whether a given vertex is min-forced is co-NP-hard (Junnila et al., 1 Sep 2025).

An analogous notion for metric dimension is the basis forced vertex: a vertex contained in every metric basis (Hakanen et al., 2021). Here the structure is different. No cut-vertex of a finite graph is basis forced, and trees have no basis forced vertices (Hakanen et al., 2021). If a graph has $S$9 vertices and $F(G)$0 basis forced vertices, then

$F(G)$1

(Hakanen et al., 2021). Moreover, if $F(G)$2 is connected with $F(G)$3 and has at least one basis forced vertex, then

$F(G)$4

(Hakanen et al., 2021). Thus basis forced vertices are rare in both sparse and dense regimes.

Paths illustrate how these notions can bifurcate. For locating-dominating codes, the number $F(G)$5 of minimum locating-dominating codes in $F(G)$6 is given explicitly, and $F(G)$7, so paths of order divisible by $F(G)$8 have a unique minimum locating-dominating code (Junnila et al., 1 Sep 2025). By contrast, for metric bases, paths do not exhibit basis forced vertices in the nontrivial cases summarized in the paper (Hakanen et al., 2021).

6. Minimal versus minimum, excluded vertices, and algorithmic boundaries

A nearby but distinct theme concerns vertices that belong to no minimal solution. In zero forcing, a graph is well-forced if all minimal zero forcing sets are minimum size (Grood et al., 2023). For trees, this property is completely characterized: a tree is well-forced if and only if repeated star removals end in a forest each of whose components is $F(G)$9 (Grood et al., 2023). The same paper characterizes irrelevant vertices in trees: a vertex belongs to no minimal zero forcing set if and only if it is a $\Pi \subseteq 2^E$00-vertex, obtained by iteratively removing vertices with double pendants and their pseudoleaves (Grood et al., 2023). This is the “forced-out” counterpart of min-forcedness.

Minimal feedback vertex sets bring another angle. Max Min FVS asks whether there exists a minimal feedback vertex set of size at least $\Pi \subseteq 2^E$01 (Lampis et al., 2023). Minimality is expressed by private cycles: a feedback vertex set $\Pi \subseteq 2^E$02 is minimal if and only if for every $\Pi \subseteq 2^E$03, the graph $\Pi \subseteq 2^E$04 contains a cycle (Lampis et al., 2023). The problem admits a treewidth dynamic program running in time

$\Pi \subseteq 2^E$05

and a branching algorithm running in time

$\Pi \subseteq 2^E$06

(Lampis et al., 2023). At the same time, ETH-based lower bounds rule out $\Pi \subseteq 2^E$07 and therefore $\Pi \subseteq 2^E$08 algorithms unless ETH fails (Lampis et al., 2023). The paper also uses explicit force gadgets to ensure that certain vertices cannot appear in any sufficiently large minimal feedback vertex set (Lampis et al., 2023). This suggests a parameterized route to vertex-forcedness questions for large minimal FVSs, even though the paper itself studies existence rather than unavoidable membership.

Across these settings, two recurrent boundaries appear. First, local gadgetry often enforces or forbids membership in minimum structures. Second, recognition is usually hard: min-forced vertices for locating-dominating codes are co-NP-hard to detect (Junnila et al., 1 Sep 2025), basis forced vertices for metric bases are co-NP-hard to detect and void vertices are NP-hard to detect (Hakanen et al., 2021). The broad picture is therefore structural rather than purely algorithmic: min-forced vertices encode indispensable roles in optimal solutions, but those roles are typically global, sparse, and computationally delicate.