On Lower Orientable Strong Diameter and Strong Radius of Some Graphs

Abstract

For two vertices $u$ and $v$ in a strong digraph $D$, the strong distance between $u$ and $v$ is the minimum number of arcs of a strong subdigraph of $D$ containing $u$ and $v$. The strong eccentricity of a vertex $v$ of $D$ is the strong distance between $v$ and a vertex farthest from $v$. The strong diameter (strong radius) of $D$ is the maximum (minimum) strong eccentricity among all vertices of $D$. The lower orientable strong diameter (lower orientable strong radius), $\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)$ ($\mathrm{s}\mathrm{r}\mathrm{a}\mathrm{d}(G)$), of a 2-edge-connected graph $G$ is the minimum strong diameter (minimum strong radius) over all strong orientations of $G$. In this paper, a conjecture of Chen and Guo is disproved by proving $\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(K_3\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_3)=\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(K_3\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_4)=5$, $\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(K_m\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{P}_n)$ is determined, $\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)$ and $\mathrm{s}\mathrm{r}\mathrm{a}\mathrm{d}(G)$ for cycle vertex multiplications are computed, and some results concerning $\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)$ are described.