English

Variety of mutual-visibility problems in hypercubes

Combinatorics 2024-05-10 v1

Abstract

Let GG be a graph and MV(G)M \subseteq V(G). Vertices x,yMx, y \in M are MM-visible if there exists a shortest x,yx,y-path of GG that does not pass through any vertex of M{x,y}M \setminus \{x, y \}. We say that MM is a mutual-visibility set if each pair of vertices of MM is MM-visible, while the size of any largest mutual-visibility set of GG is the mutual-visibility number of GG. If some additional combinations for pairs of vertices x,yx, y are required to be MM-visible, we obtain the total (every x,yV(G)x,y \in V(G) are MM-visible), the outer (every xMx \in M and every yV(G)My \in V(G) \setminus M are MM-visible), and the dual (every x,yV(G)Mx,y \in V(G) \setminus M are MM-visible) mutual-visibility set of GG. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of GG, respectively. We present results on the variety of mutual-visibility problems in hypercubes.

Cite

@article{arxiv.2405.05650,
  title  = {Variety of mutual-visibility problems in hypercubes},
  author = {Danilo Korže and Aleksander Vesel},
  journal= {arXiv preprint arXiv:2405.05650},
  year   = {2024}
}
R2 v1 2026-06-28T16:21:53.956Z