Mutual-Visibility Sets in Graph Theory

• Mutual-visibility sets are sets of vertices in a graph where every pair is connected by a shortest path that avoids all other selected vertices.
• The concept extends to variants such as total, outer, and dual visibility, incorporating convexity properties and polynomial invariants to refine analysis.
• Algorithmic and complexity studies show NP-completeness in general while enabling efficient solutions on structured classes, underlining the practical significance of these sets.

Mutual-visibility sets are geodesic packing objects in graphs. For a graph $G=(V(G),E(G))$ and a set $X\subseteq V(G)$, two vertices $u,v\in V(G)$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that

$V(P)\cap X\subseteq \{u,v\}.$

A set $X$ is a mutual-visibility set if every two vertices of $X$ are $X$-visible, and the maximum size of such a set is the mutual-visibility number

$X\subseteq V(G)$0

The notion was introduced as a graph-theoretic geodesic analogue of visibility and of the condition that no third selected point lies on a shortest segment, and it has since developed into a broader theory involving stronger visibility variants, exact formulas on structured graph classes, polynomial invariants, approximation and hardness results, product constructions, and directed extensions (Stefano, 2021, Cicerone et al., 2023, Stojanović, 3 Feb 2026).

1. Foundational definitions and geodesic viewpoint

The defining feature of mutual visibility is existential geodesic avoidance: for $X\subseteq V(G)$1, it is enough that some shortest $X\subseteq V(G)$2-path avoids $X\subseteq V(G)$3. This distinguishes mutual visibility from the general position condition, where every geodesic between two selected vertices must avoid the rest of the set. Every general position set is a mutual-visibility set, but not conversely; the relation “mutually visible” on a fixed set is reflexive and symmetric, but not transitive (Stefano, 2021).

A useful structural convention is convexity. If $X\subseteq V(G)$4 is a convex subgraph of $X\subseteq V(G)$5, then restriction preserves mutual visibility: if $X\subseteq V(G)$6 is a mutual-visibility set of $X\subseteq V(G)$7, then $X\subseteq V(G)$8 is a mutual-visibility set of $X\subseteq V(G)$9. Consequently,

$u,v\in V(G)$0

for every convex subgraph $u,v\in V(G)$1 of $u,v\in V(G)$2. Ordinary induced subgraphs do not preserve the parameter in general, which is why convexity, rather than inducedness, is the natural hereditary notion in the subject (Stefano, 2021).

Several extremal values were identified at the outset. For connected graphs,

$u,v\in V(G)$3

and

$u,v\in V(G)$4

if and only if the graph is a path $u,v\in V(G)$5 or, in the disconnected setting, a disjoint union of path graphs. Complete graphs are exactly those with $u,v\in V(G)$6. These observations already show that mutual visibility is neither a purely local nor a purely density-driven invariant: it is controlled by how shortest paths thread through selected vertices (Stefano, 2021).

2. Variants, spectra, and coloring refinements

A major extension replaces “pairs inside $u,v\in V(G)$7” by other natural classes of vertex pairs. Writing $u,v\in V(G)$8, three standard variants are defined (Cicerone et al., 2023):

Notion	Required $u,v\in V(G)$9-visible pairs	Maximum size
Mutual	all $X$0	$X$1
Total	all $X$2	$X$3
Outer	all $X$4, and all $X$5	$X$6
Dual	all $X$7, and all $X$8	$X$9

These parameters satisfy the basic inequalities

$u,v$0

Mutual-, outer-, and total mutual-visibility sets are downward closed under taking subsets, but dual mutual-visibility sets are not; $u,v$1 gives a concrete example, where an edge is a dual mutual-visibility set but a singleton need not be (Cicerone et al., 2023).

The subject also has an enumerative side. The visibility polynomial of a graph is

$u,v$2

where $u,v$3 counts mutual-visibility sets of size $u,v$4. Analogous dual, outer, and total visibility polynomials are defined. Because mutual-, outer-, and total visibility families are closed under subsets, their feasible sizes form initial intervals $u,v$5, $u,v$6, and $u,v$7. Dual visibility behaves differently: for every finite subset $u,v$8 of positive integers there exists a graph $u,v$9 that has a dual mutual-visibility set of size $P$0 if and only if $P$1, and every polynomial with nonnegative integer coefficients and constant term $P$2 is a dual visibility polynomial of some graph (Bujtás et al., 2024).

A further refinement adds independence. An independent mutual-visibility set is an independent set that is also a mutual-visibility set; its maximum size is $P$3. The corresponding chromatic parameter $P$4 is the minimum number of colors in a vertex partition into independent mutual-visibility sets, while $P$5 is the minimum number of colors in a partition into mutual-visibility sets. These parameters connect visibility theory to ordinary coloring, independence, defective coloring, and Ramsey theory. In particular, if $P$6, then

$P$7

while for trees of order at least $P$8,

$P$9

(Brešar et al., 7 May 2025).

3. Structural characterizations

A central theme of the theory is that visibility constraints can often be reformulated as local geodesic obstructions. For total mutual visibility, the decisive characterization is that $V(P)\cap X\subseteq \{u,v\}.$0 is a total mutual-visibility set if and only if every pair $V(P)\cap X\subseteq \{u,v\}.$1 with $V(P)\cap X\subseteq \{u,v\}.$2 is $V(P)\cap X\subseteq \{u,v\}.$3-visible, equivalently if and only if every distance-$V(P)\cap X\subseteq \{u,v\}.$4 pair satisfies

$V(P)\cap X\subseteq \{u,v\}.$5

This converts a global all-pairs definition into a local common-neighbor condition. In geodetic graphs, the total mutual-visibility number equals the number of simplicial vertices, and the simplicial set is the unique maximum total mutual-visibility set (Bujtás et al., 2024).

The vanishing regimes of the stronger variants are also well developed. For total mutual visibility,

$V(P)\cap X\subseteq \{u,v\}.$6

for a graph $V(P)\cap X\subseteq \{u,v\}.$7 with $V(P)\cap X\subseteq \{u,v\}.$8 if and only if each vertex is the middle vertex of a convex $V(P)\cap X\subseteq \{u,v\}.$9. For dual mutual visibility, if every two adjacent vertices are the center of a convex $X$0, then

$X$1

and if $X$2 has girth at least $X$3, then

$X$4

A convex-cover criterion further shows that if $X$5 is covered by convex induced subgraphs each having dual mutual-visibility number $X$6, then $X$7 (Cicerone et al., 2023).

Trees admit a complete intrinsic characterization. A set $X$8 is a mutual-visibility set of a tree $X$9 if and only if it is exactly the leaf set of the Steiner subtree $X$0: $X$1 As a consequence,

$X$2

If $X$3 has branch vertices and $X$4 denotes the length of the leg determined by a leaf $X$5, then the number of maximum mutual-visibility sets is

$X$6

The same work proves that every tree is absolute-clear and that for trees with at least two edges,

$X$7

(B et al., 13 Jan 2026).

For distance-hereditary graphs, the structure is described by the canonical split decomposition. In that setting the only essential obstructions are the centers of certain star bags, called $X$8-vertices, and terminal arrows in a directed version of the decomposition. This yields a linear-time algorithm that computes an optimal mutual-visibility set, so the NP-complete general problem becomes tractable on this decomposition-defined class (Cicerone et al., 2023).

4. Exact formulas on major graph classes

The original paper established exact values on several basic classes. For connected block graphs, if $X$9 is the set of articulation vertices, then $X$0 is a mutual-visibility set and

$X$1

For trees this specializes to the leaf formula above. For complete bipartite graphs,

$X$2

and for connected cographs,

$X$3

with a polynomial-time construction of a maximum mutual-visibility set (Stefano, 2021).

Cycles are the smallest family where the four standard parameters separate sharply. For $X$4,

$X$5

while

$X$6

and

$X$7

These formulas already show that the classical, total, outer, and dual notions are genuinely different invariants (Cicerone et al., 2023).

For Cartesian grids,

$X$8

For higher-dimensional grids

$X$9

the total mutual-visibility number is

$X\subseteq V(G)$00

In two dimensions, the outer and dual parameters are known exactly; for large grids,

$X\subseteq V(G)$01

when $X\subseteq V(G)$02. This provides a concrete family where all four parameters differ and the gaps can be arbitrarily large (Cicerone et al., 2023).

Tori show a complementary pattern. For the ordinary mutual-visibility number, the general upper bound

$X\subseteq V(G)$03

was established first, and later work proved that for $X\subseteq V(G)$04 and $X\subseteq V(G)$05 or $X\subseteq V(G)$06,

$X\subseteq V(G)$07

For cylinders,

$X\subseteq V(G)$08

whenever $X\subseteq V(G)$09, and a conjecture states that for $X\subseteq V(G)$10,

$X\subseteq V(G)$11

(Stefano, 2021, Korže et al., 2023). For the stronger variants on tori,

$X\subseteq V(G)$12

and

$X\subseteq V(G)$13

while the exact value of $X\subseteq V(G)$14 remains open; the known bound is

$X\subseteq V(G)$15

for $X\subseteq V(G)$16 (Cicerone et al., 2023).

Self-similar families produce further exact theories. For Sierpiński triangle graphs $X\subseteq V(G)$17, with the notation $X\subseteq V(G)$18 in that paper corresponding to $X\subseteq V(G)$19,

$X\subseteq V(G)$20

$X\subseteq V(G)$21

$X\subseteq V(G)$22

Here the optimal ordinary mutual-visibility and general position sets coincide for all $X\subseteq V(G)$23, while the stronger visibility variants collapse to much smaller values (Korže et al., 2024).

Hypercube-like graphs add both exact and asymptotic results. For hypercubes,

$X\subseteq V(G)$24

so

$X\subseteq V(G)$25

and the total mutual-visibility number satisfies asymptotically tight bounds

$X\subseteq V(G)$26

For cube-connected cycles,

$X\subseteq V(G)$27

and for butterflies,

$X\subseteq V(G)$28

(Cicerone et al., 2023, Axenovich et al., 2024).

5. Algorithms, verification, and computational complexity

The decision version of the classical problem was shown to be NP-complete in the foundational paper via a reduction from $X\subseteq V(G)$29-SAT, while verification is polynomial-time. Given $X\subseteq V(G)$30 and $X\subseteq V(G)$31, one can test whether $X\subseteq V(G)$32 is a mutual-visibility set in

$X\subseteq V(G)$33

time by comparing shortest-path distances in $X\subseteq V(G)$34 and in graphs obtained by deleting the other selected vertices (Stefano, 2021).

The later “variety” paper generalized this picture to all four parameters. For any

$X\subseteq V(G)$35

the corresponding decision problem “is $X\subseteq V(G)$36?” is NP-complete. Membership in NP again comes from polynomial-time verification, with the explicit bounds:

• mutual-, outer-, and total mutual-visibility verification in $X\subseteq V(G)$37,
• dual mutual-visibility verification in $X\subseteq V(G)$38. The NP-completeness proof is a simultaneous reduction from Independent Set showing

$X\subseteq V(G)$39

for a suitable graph $X\subseteq V(G)$40 (Cicerone et al., 2023).

Approximation is also hard. A polynomial-time algorithm finds a mutual-visibility set of size

$X\subseteq V(G)$41

where $X\subseteq V(G)$42 is the average distance in the graph. On the negative side, for graphs of diameter at least $X\subseteq V(G)$43 and every constant $X\subseteq V(G)$44, mutual-visibility and dual mutual-visibility are not approximable within a factor of

$X\subseteq V(G)$45

while outer and total mutual-visibility are not approximable within

$X\subseteq V(G)$46

unless $X\subseteq V(G)$47. All four visibility problems are also $X\subseteq V(G)$48-hard on graphs of diameter $X\subseteq V(G)$49 (Bilò et al., 2024).

Despite these hardness results, some graph classes admit exact fast algorithms. The most prominent example is distance-hereditary graphs, where canonical split decomposition yields a linear-time algorithm computing an optimal mutual-visibility set. The algorithm removes all $X\subseteq V(G)$50-vertices, identifies terminal arrows in the directed decomposition, and performs a minimal case-dependent set of deletions from the initial candidate $X\subseteq V(G)$51 (Cicerone et al., 2023).

The independent and coloring variants remain computationally rich. Computing $X\subseteq V(G)$52 and $X\subseteq V(G)$53 is NP-complete even for graphs with universal vertices, and it is NP-hard to decide whether

$X\subseteq V(G)$54

For ordinary visibility coloring, NP-completeness remains a conjecture (Brešar et al., 7 May 2025).

6. Products, perturbations, directed graphs, and open directions

Graph products are a recurrent source of large, explicitly describable mutual-visibility sets. For strong products, a paper centered on total mutual-visibility introduced lower bounds of the form

$X\subseteq V(G)$55

and, using feasible total mutual-visibility sets, sharper product bounds leading to an exact formula for strong grids: $X\subseteq V(G)$56 For strong prisms over block graphs,

$X\subseteq V(G)$57

(Cicerone et al., 2022).

For corona products, the geometry is even more rigid. If $X\subseteq V(G)$58 and $X\subseteq V(G)$59 each have at least two vertices, then

$X\subseteq V(G)$60

The maximum set is exactly the union of all pendant copies of $X\subseteq V(G)$61, excluding the core vertices of $X\subseteq V(G)$62. This structure extends to counting formulas for the visibility polynomial of $X\subseteq V(G)$63, via the auxiliary notion of $X\subseteq V(G)$64-visible sets and maximal absolute $X\subseteq V(G)$65-visible sets (B et al., 2 Sep 2025).

The theory is no longer confined to undirected graphs. In directed graphs, a set $X\subseteq V(G)$66 is mutual-visible only if for every $X\subseteq V(G)$67 there exist shortest directed paths from $X\subseteq V(G)$68 to $X\subseteq V(G)$69 and from $X\subseteq V(G)$70 to $X\subseteq V(G)$71, both avoiding $X\subseteq V(G)$72 internally. This immediately forces nontrivial sets to lie in a single strongly connected component, and yields

$X\subseteq V(G)$73

For directed acyclic graphs,

$X\subseteq V(G)$74

for directed cycles of length at least $X\subseteq V(G)$75,

$X\subseteq V(G)$76

while tournaments can have arbitrarily large mutual-visibility sets; for Paley tournaments $X\subseteq V(G)$77,

$X\subseteq V(G)$78

Verification remains polynomial-time solvable, but computing $X\subseteq V(G)$79 is NP-hard (Stojanović, 3 Feb 2026).

Local graph modifications reveal a strong instability phenomenon. If $X\subseteq V(G)$80 remains connected, then

$X\subseteq V(G)$81

and

$X\subseteq V(G)$82

If $X\subseteq V(G)$83 remains connected, then

$X\subseteq V(G)$84

Beyond these bounds, most remaining edge- and vertex-deletion cases are highly unstable; for example, neither $X\subseteq V(G)$85 nor $X\subseteq V(G)$86 admits a universal affine bound of the form $X\subseteq V(G)$87 with $X\subseteq V(G)$88 (Dokyeesun et al., 25 May 2025).

Several open problems remain central. The exact value of $X\subseteq V(G)$89 was historically open for tori and is now known in large ranges, but the outer mutual-visibility number of tori is still unresolved (Stefano, 2021, Cicerone et al., 2023). The cylinder formula

$X\subseteq V(G)$90

is conjectural outside the proven range $X\subseteq V(G)$91 (Korže et al., 2023). For visibility colorings, NP-completeness of the ordinary MV coloring problem is conjectured (Brešar et al., 7 May 2025). More broadly, the field continues to branch toward visibility polynomials, graph products, directed and weighted settings, and finer stability questions under local modifications (B et al., 2 Jul 2025, Bujtás et al., 2024).