On the mutual visibility in Cartesian products and triangle-free graphs

Injective coloring of graphs revisited

“…and only if S does not contain a subgraph isomorphic to C 4 (this fact was proved in [7]). Equivalently, this means that the subgraph of B induced by the edges in B that correspond to the vertices of S in K m K n do not contain K 2,2 as a subgraph.…”

“…The problem of mutual-visibility in Cartesian products of two complete graphs is intrinsically related to the famous Zarankiewicz problem, which is still open. More precisely, it was noticed by Cicerone, Di Stefano and Klavžar in [7] that µ(K m K n ) equals z(m, n; 2, 2), which is the maximum number of 1s in an m × n binary matrix that contains no constant 2 × 2 submatrix of 1s; see [7] for more details. Here we present a similarly strong connection between the lower mutual-visibility number of Cartesian products of two complete graphs with another old result related to binary matrices.…”

“…The concept of mutual visibility was recently introduced by Di Stefano in [9] with a primary motivation to achieve confidential communication between mobile entities in a network; see [9] for further references containing several other earlier studies. The paper was followed by more studies that all arose within the last year [6][7][8]19]. Given a graph G and X ⊂ V (G), two vertices a, b ∈ V (G) are X-visible if there exists a shortest a, b-path P in G such that V (P ) ∩ X ⊆ {a, b}.…”

“…Given a graph G and X ⊂ V (G), two vertices a, b ∈ V (G) are X-visible if there exists a shortest a, b-path P in G such that V (P ) ∩ X ⊆ {a, b}. If each two vertices in X are X-visible, then X is a mutual-visibility set of G. The cardinality of a largest mutual-visibility set in G is the mutual-visibility number, µ(G), of G. While studying mutual-visibility in strong products of graphs, the authors of [8] encountered the following useful and natural variation. A set X of vertices in G is a total mutualvisibility set if every two vertices in G (not only in X!)…”

“…We initiate here the study of two variations of the two concepts above, which are inline with the initial discussion on upper/lower versions of graph invariants. In particular, since the decision problems of determining µ(G) and µ t (G) are NPcomplete [6], it is interesting to obtain lower bounds and potential approximations of the studied invariants. A set X ⊂ V (G) is a maximal (total) mutual-visibility set in G if X is a (total) mutual visibility set in G and every set Y , with X Y , is not a (total) mutual-visibility set in G. The cardinality of a smallest maximal (total) mutual-visibility set is the lower (total) mutual-visibility number of G, denoted by µ − (G), respectively µ − t (G).…”