17 problems

For positive integers $t$ and $r$, consider optimal $(t,r)$ broadcasts on the infinite grid and their densities. Blessing et al.'s conjecture. The optimal $(t,r)$ and $(t+1,r+2)$ b…

Let $H_{m,m}$ be the finite truncated square tiling graph with $m\geq 1$, let $\gamma_{t,r}(H_{m,m})$ denote its $(t,r)$ broadcast domination number, and let $m$ and $r$ be positiv…

Let $H_{\infty,\infty}$ be the infinite truncated square tiling graph, and let $\delta_{t,r}(H_{\infty,\infty})$ denote its optimal $(t,r)$ broadcast density. Density conjecture. T…

Full directed broadcast domination interval conjecture. For every $b\in\mathbb Z\cap[d,D]$, there exists an orientation $\vec G_b$ of $G$ satisfying

Full-interval conjecture. For every integer $b\in\mathbb Z\cap[d,D]$, there exists an orientation $\vec G_b$ of $G$ such that

In-degree characterization conjecture. A vertex is a dominating vertex if and only if it has in-degree at most $1$; consequently, the orientations described above that maximize the…

Let a set of towers $\mathcal{T}\subset\mathbb{R}^2$ be $(t,r)$-broadcasting if every point $v\in\mathbb{R}^2$ satisfies … where $d$ is the $\ell_1$ distance. Let…

Let $t$ and $r$ be positive integers, and let $\delta_{t,r}(\mathbb{Z}^2)$ denote the minimum density of a $(t,r)$-broadcasting set in the infinite square grid. Asymptotic sharpnes…

Let $t,r$ be positive integers, and let $\delta_{t,r}(\mathbb{Z}^2)$ denote the minimum density of a $(t,r)$-broadcasting set in the infinite square grid. The preceding bounds comp…

Let $\gamma_{t,r}(G)$ be the $(t,r)$ broadcast domination number of a finite graph $G$. For finite graphs $G$ and $H$, the half-factor broadcast-domination conjecture. … This conje…

For a finite graph $G$, let $\gamma_{t,r}(G)$ denote its $(t,r)$ broadcast domination number. The paper has shown that the analogous bound fails for $r>1$, while its counterexample…

Let $G_\infty$ be the infinite grid, and let $\delta_{t,r}(G_\infty)$ denote the minimum density of a $(t,r)$ broadcast. Revised broadcast-density conjecture. For all $t>2$, … This…

Uniqueness conjecture. The trees in that proposition are the only trees reaching the bound.

Let $G$ be a graph. An irredundant broadcast is a broadcast $f$; write $\operatorname{ir}_{b}(G)$ for the minimum cost of an irredundant broadcast, $\gamma_{b}(G)$ for the minimum…

Parameter-shift conjecture. The optimal $(t,r)$ and $(t-1,r-2)$ broadcast dominating sets of ${\mathbb Z}\times{\mathbb Z}$ are equal.

Large-grid tightness conjecture. The bounds are tight when $n\geq m>3p$ under this homomorphism-preimage condition.

Optimality conjecture. When $m$ and $n$ are sufficiently large, this constructed set is an optimal dominating set of $G_{m,n}$.