Mutual-visibility problems on graphs of diameter two

TL;DR

The work advances the theory of mutual-visibility in diameter-two graphs by deriving exact and asymptotic formulas for the four invariants $\mu$, $\mu_d$, $\mu_o$, and $\mu_t$ across key graph families, notably Cartesian and direct products of complete graphs, their line graphs, and cographs, and by analyzing extremal diameter-two graphs of minimum size. It reveals deep connections to classical extremal problems, showing, for example, $\mu_d(K_n\square K_m)=n+m-1$, $\mu_o(K_n\square K_m)=n+m-2$ for $n,m\ge3$, and that for $K_n\times K_m$ with $n,m\ge5$ all four invariants coincide at $nm-4$; in line graphs, $\mu(L(K_n))=z(n,n;2,2)$ and $\mu_t(L(K_n))=\mathrm{ex}(n; C_4)$, with analogous results for $L(K_{m,n})$. The paper also provides a complete characterization of the mutual-visibility landscape on cographs and exact formulas for the extremal diameter-two graphs in the Erdős–Rényi/Henning–Southey family, illustrating how duplicating degree-2 vertices affects each invariant. These results open routes to complexity considerations and to broader explorations of diameter-two line graphs and forbidden-subgraph extremals.

Abstract

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.

Figures (4)

• Figure 1: Cases from the proof of Theorem \ref{['thm:hamming-outer-dual']}(i)
• Figure 2: Forbidden induced subgraphs for $(K_n)_F$, where $F$ is such that $S_F$ is a (outer, dual, total) mutual visibility set of $L(K_n))$
• Figure 3: Some examples of graphs from the family $\mathcal{G}$.
• Figure 4: Variations of mutual-visibility sets in some graphs of the family $\mathcal{G}$.

Theorems & Definitions (41)

• Lemma 1.1
• Theorem 2.1
• Theorem 2.2
• Corollary 2.3
• Lemma 3.1
• Theorem 3.2
• Corollary 3.3
• Lemma 3.4
• Proposition 3.5
• Definition 3.6