General position and mutual-visibility in shadow graphs
Haritha S., Ullas Chandran S.
TL;DR
This paper investigates general position and mutual-visibility numbers in shadow graphs $S(G)$. It develops structural tools via the gp-partition and distance lemmas to derive tight bounds and exact values for $gp(S(G))$ across classes such as complete graphs, complete bipartite graphs, cycles, and trees, with key results including $gp(S(K_n))=n$, $gp(S(K_{m,n}))=2m$, and $gp(S(T))=2\,l(T)$ for trees. It then provides tight bounds and characterizations for the mutual-visibility number $\mu(S(G))$, showing $\max\{n(G),2\,\mu_i(G),2\,\Delta(G)\}\le \mu(S(G)) \le \min\{n(G)+\mu(G),\,2n(G)-2\}$ and giving exact values for several graph families (e.g., cycles, complete multipartite graphs, and trees) along with a complete characterization for the smallest possible nontrivial values $\mu(S(G))=2$ or $4$ and the impossibility of $\mu(S(G))\in\{3,5\}$ in connected graphs.
Abstract
The \emph{general position problem} in graphs asks for a largest set of vertices in which no three lie on a common shortest path. The \emph{mutual-visibility problem} seeks a largest set of vertices such that every pair is connected by a shortest path whose internal vertices lie outside the set. In this paper, we investigate the general position and mutual-visibility problems for shadow graphs. Sharp general bounds are established for both the general position number and the mutual-visibility number of shadow graphs, and classes of graphs attaining these extremal values are characterized. Furthermore, these invariants are determined for several standard classes of shadow graphs, including shadow graphs of cycles, multipartite graphs, and trees.