2024/05/09 by Danilo Korže, Korže, Danilo, Aleksander Vesel +1 · 6 citations
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #Optimization and Search Problems #Software-Defined Networks and 5G
paper · pdf · doi:10.48550/arxiv.2405.05650
openalex publication_date 2024/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a graph and M ⊆ V(G). Vertices x, y ∈ M are M-visible if there exists a shortest x,y-path of G that does not pass through any vertex of M ∖ \x, y \. We say that M is a mutual-visibility set if each pair of vertices of M is M-visible, while the size of any largest mutual-visibility set of G is the mutual-visibility number of G. If some additional combinations for pairs of vertices x, y are required to be M-visible, we obtain the total (every x,y ∈ V(G) are M-visible), the outer (every x ∈ M and every y ∈ V(G) ∖ M are M-visible), and the dual (every x,y ∈ V(G) ∖ M are M-visible) mutual-visibility set of G. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of G, respectively. We present results on the variety of mutual-visibility problems in hypercubes.