---
title: Mutual-Visibility Chromatic Number
url: https://www.emergentmind.com/topics/mutual-visibility-chromatic-number
type: topic
---

# Mutual-Visibility Chromatic Number

The mutual-visibility chromatic number, denoted $\chi_\mu(G)$, is an invariant of finite graphs that quantifies the minimum number of colors required to partition the vertex set into classes in which every pair of vertices is connected by a shortest path whose internal vertices are not assigned that color. This property generalizes classical vertex coloring by replacing independence with geodesic visibility within each color class. The recent foundational works [2512.12251], [2505.04144], and [2408.03132] have established its formal definition, complexity, extremal bounds, exact formulas for families of graphs, and links to other combinatorial parameters.

## 1. Precise Definition and Core Properties

Given a connected graph $G=(V,E)$, a subset $S \subseteq V$ is a mutual-visibility set if for any pair $x, y \in S$, there exists a shortest $x$–$y$ path whose internal vertices all lie outside $S$. Formally, a coloring $c:V \to [k]$ is a mutual-visibility coloring if for each color $i$, the set $S_i = c^{-1}(i)$ satisfies: for all $x, y \in S_i$, there exists a shortest $x$–$y$ path $P$ in $G$ such that $\operatorname{Int}(P) \cap S_i = \emptyset$ [2512.12251]. The mutual-visibility chromatic number is thus
\[
\chi_\mu(G) = \min\{ k \mid V(G) \text{ admits a partition into } k \text{ mutual-visibility sets} \}.
\]

For geodetic graphs (graphs with unique shortest paths), at most two vertices per color are possible on a diameter path, yielding $\chi_\mu(G) \ge \lceil(\mathrm{diam}(G)+1)/2\rceil$ [2408.03132].

## 2. Computational Complexity and NP-Completeness

The decision problem for mutual-visibility coloring is to determine, for given $G$ and $k$, whether $\chi_\mu(G) \leq k$. This problem is NP-complete for $k=2$ even when restricted to graphs of diameter four [2512.12251]. The reduction from NAE-3SAT utilizes gadgets (notably the $H_n$ construction) that enforce two mutually visible color classes whose internal structure mirrors the logical constraints of NAE-3SAT. This shows the nontriviality of the constraint and the impossibility of efficient algorithms for general input unless P = NP.

The related independent mutual-visibility chromatic number $\chi_{\mu_i}(G)$—where each color class is both independent and mutual-visible—is also NP-complete to compute [2505.04144]. In addition, deciding equality $\imv(G) = \alpha(G)$ is NP-hard, establishing the intrinsic complexity even in restricted graph families.

## 3. Exact Formulas and Extremal Bounds

Numerous sharp bounds and explicit formulas have been established:

| Graph Family           | $\chi_\mu(G)$              | Reference         |
|------------------------|----------------------------|-------------------|
| Path $P_n$             | $\lceil(n+1)/2\rceil$      | [2408.03132], [2505.04144] |
| Cycle $C_n$ ($n\ge 6$) | $\lfloor n/3 \rfloor$      | [2505.04144]      |
| Tree $T$               | $\operatorname{rad}(T)+1$ if even diameter; $\operatorname{rad}(T)$ if odd | [2408.03132] |
| Glued binary trees $GT(r)$ | Piecewise: $2(r-i)+3$ or $2(r-i)+2$ for integer $i$ depending on $r$ | [2512.12251] |
| Block graph, $\text{diam}=d$ | $\lceil(d+1)/2\rceil$ | [2408.03132], [2510.10284] |

The extremal lower bound $\chi_\mu(G) \ge \lceil |V(G)|/\mu(G)\rceil$ holds for general graphs ($\mu(G)$ is the maximal mutual-visibility set) [2408.03132].

For Cartesian products of cliques, $\chi_\mu(K_n \square K_n) = \Theta(\sqrt{n})$ [2408.03132]. For certain highly symmetric constructions—e.g., glued $t$-ary trees—the parameter is described by explicit piecewise formulas with transitions determined by the relationship between depth and arity [2512.12251].

## 4. Graph Products and Structural Behavior

Mutual-visibility coloring interacts richly with graph products:

- For the lexicographic product $G \circ H$, if at least one factor is non-complete and $G$ is connected with $|G|\ge 2$, then $\chi_\mu(G \circ H) = 2$ [2505.04144].
- For strong products and Cartesian products, sharp lower and upper bounds exist, often involving domination parameters or packing numbers [2510.10284, 2505.04144]:
  \[
  \chi_{\mu_2}(G \square H) \ge \max\{\chi_{\mu_2}(G) \rho_2(H), \chi_{\mu_2}(H) \rho_2(G)\}.
  \]
  For strong products, $\chi_{\mu_k}(G \boxtimes H) \leq \chi_{\mu_k}(G)\chi_{\mu_k}(H)$.

In Hamming graphs (products of complete graphs), determining $\chi_\mu$ is open beyond specific cases such as Cartesian products of two cliques [2408.03132]. For hypercubes $Q_n$, it is verified that
\[
\chi_\mu(Q_n) = \Theta(\log\log n),
\]
disproving conjectures that it may be bounded by a constant [2402.04791].

## 5. Algorithmic Aspects and Related Parameters

Finding a maximum mutual-visibility set is NP-hard, hence greedy coloring algorithms may fail to achieve optimal colorings in general [2408.03132]. However, specialized structures—block graphs, strong grids, certain product graphs—admit efficient and sometimes optimal greedy solutions.

The mutual-visibility chromatic number is bounded above by several related parameters: in graphs of diameter two, $\chi_\mu(G) \leq \chi(G)$, and more generally, for regular graphs of large girth,
\[
\chi_\mu(G) \leq \lceil (n(G) - r^2 + 4)/2 \rceil
\]
holds for $r$-regular $G$ with girth $>6$ [2408.03132].

In the $k$-distance mutual-visibility generalization ($\chi_{\mu_k}(G)$), the classical clique-cover ($k=1$) and mutual-visibility chromatic number ($k \geq \text{diam}(G)$) appear as extremal cases [2510.10284]. For $k=2$, tight bounds in terms of domination and total domination numbers have been established.

## 6. Relation to General Position and Ramsey Theory

The mutual-visibility chromatic number connects structurally to other combinatorial invariants, e.g.:
- General position chromatic number $\chi_{gp}(G)$.
- Domino parameters: domination number $\gamma(G)$, total domination number $\gamma_t(G)$.
- In subdivisions of complete graphs $S(K_n)$, $\chi_\mu(S(K_n))$ is tightly linked to Ramsey numbers $R(4;k)$: the minimum number $p(n)$ so that every edge-coloring of $K_n$ with $p(n)$ colors has a monochromatic $K_4$ [2505.04144]. This demonstrates Ramsey-theoretic intractability in exact computation.

In exact-distance graphs and corona constructions, new bounds and equivalences have emerged, notably $\chi_\mu(G) \leq \chi_\mu(G \circ H) \leq \chi_\mu(G) + 1$ [2408.03132].

## 7. Open Questions and Future Research Directions

Several open problems are fundamental:
- Determining $\chi_\mu(G)$ for families such as hypercubes, Hamming graphs, Sierpiński graphs, and product graphs in general [2512.12251, 2402.04791, 2408.03132].
- Full characterization of graphs with $\chi_\mu(G)=2$ [2408.03132].
- Determining the computational complexity in broader settings, with multiple conjectures regarding NP-completeness [2505.04144].
- Study extremal bounds and efficient approximation algorithms in sparse graph classes (planar, chordal) and links to zero-forcing and general position numbers [2512.12251].

The interplay of mutual-visibility colorings with graph products, domination, partitioning, and Ramsey theory marks this parameter as central to emerging research in combinatorial optimization, extremal graph theory, and algorithmic complexity.

Source: https://www.emergentmind.com/topics/mutual-visibility-chromatic-number