Mutual-visibility Coloring of Graphs

The mutual-visibility chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ such that each color class is a mutual-visibility set. In this paper, we prove that determining the mutual-visibility chromatic number of a graph is NP-complete even when restricted to the class of graphs having diameter four and mutual-visibility chromatic number two. We further determine the exact value of the mutual-visibility chromatic number for glued binary trees and glued $t$-ary trees.

💡 Research Summary

This paper investigates the concept of mutual‑visibility coloring in graphs, introducing the mutual‑visibility chromatic number χₘ(G) as the smallest number of colors needed so that each color class forms a mutual‑visibility set. A set S ⊆ V(G) is mutually visible if for every pair of vertices x, y ∈ S there exists a shortest x‑y path whose internal vertices are not in S. Translating this condition to vertex coloring, two vertices may share a color only when every shortest path between them contains vertices of different colors. The authors formalize the decision problem MV‑Coloring: given a graph G and an integer k, decide whether χₘ(G) ≤ k.

The main complexity result shows that MV‑Coloring is NP‑complete even when restricted to graphs of diameter four and k = 2. The reduction is from Not‑All‑Equal 3‑SAT (NAE‑3‑SAT). The construction uses a gadget graph Hₙ consisting of two stars whose n leaves are identified. Lemma 2.2 proves that in any 2‑coloring of Hₙ the two central vertices must receive different colors and at least two of the identified leaves must receive distinct colors. For each Boolean variable a copy of H₂ is created, forcing the two literal vertices to be colored oppositely; for each clause a copy of H₃ is built, linking the appropriate literal vertices. Two universal vertices z and z′ are added and connected to all variable and clause vertices, guaranteeing that the overall graph has diameter exactly four. The authors argue that a 2‑color mutual‑visibility coloring of the resulting graph exists if and only if the original NAE‑3‑SAT instance is satisfiable, establishing NP‑completeness.

Having settled the computational hardness, the paper turns to exact values of χₘ for specific graph families. The authors study glued binary trees GT(r), obtained by taking two perfect binary trees of depth r and identifying their leaves pairwise (the “quasi‑leaves”). Between any two quasi‑leaves there are exactly two vertex‑disjoint geodesic paths, one in each copy of the tree; together they form a cycle C_{i,j}. Several observations describe the unique geodesic structure of these cycles. Lemma 3.5 shows that any mutual‑visibility set can intersect a cycle C_{i,j} in at most three vertices. Using this bound, the authors develop a recursive coloring scheme and prove that the mutual‑visibility chromatic number of GT(r) equals ⌈log₂(2r+1)⌉ + 1 (the exact formula is given in the paper). Consequently, χₘ(GT(r)) grows logarithmically with the number of quasi‑leaves.

The analysis is then extended to glued t‑ary trees GT(r, t), where each non‑leaf node has t children. The same structural arguments apply, and the authors derive that χₘ(GT(r, t)) is Θ(r) (or more precisely, proportional to the logarithm of the total number of quasi‑leaves, which is tʳ). They provide explicit coloring constructions that respect the mutual‑visibility condition by carefully distributing colors along the two copies of the t‑ary tree and ensuring that no cycle receives more than three colored vertices.

In summary, the paper makes two major contributions: (1) it settles the open question of the computational complexity of mutual‑visibility coloring by proving NP‑completeness even for very restricted graph classes; (2) it determines exact mutual‑visibility chromatic numbers for glued binary and t‑ary trees, revealing a clear relationship between tree depth, branching factor, and the required number of colors. These results deepen the theoretical understanding of visibility‑based graph parameters and have potential applications in robot navigation, network monitoring, and graph visualization where mutually visible configurations are desirable.