9 problems

Let $G$ and $H$ be graphs, let $G\Box H$ be their Cartesian product, and let $D$ be a dominating set of $G\Box H$. Let $M_G(A)$ denote the family of dominating sets of $G$ associat…

Let $G$ and $H$ be graphs, let $G\Box H$ be their Cartesian product, and let $D$ be a dominating set of $G\Box H$. For each $h\in V(H)$, let $G_h$ be the $G$-layer corresponding to…

Goddard–Henning conjecture. If

Let $d\geq 2$, and let $G$ be a connected bipartite graph of order $n$. Write $C_{2d+2}$ for the cycle of length $2d+2$, and let $\mathcal{B}_d$ denote the graph family defined in…

Domination conjecture for $SCC(2)$ graphs. If $G\in SCC(2)$, then

Let $S$ be a spider, meaning a tree with one vertex of degree greater than $2$, whose legs all have length at most $3$. Let $S'$ be obtained from $S$ by deleting a leaf, and let…

Let $G$ be any graph, and let $D(G)$ denote its domination polynomial, whose coefficient sequence is the sequence of numbers of dominating sets of each cardinality. Alikhani–Peng c…

Let $\mathcal{K}_n=P_n\boxtimes P_n$ be the King grid, where $P_n$ is the path on $n$ vertices. Write $\gamma^*_e(G)$ for the minimum cardinality of an exponential dominating set i…

Let $C_m$ and $C_n$ be cycle graphs, and let $C_m\square C_n$ be their Cartesian product. Write $\gamma^*_e(G)$ for the minimum cardinality of an exponential dominating set in a gr…