9 problems

Degree-bounded expected value polynomial conjecture. The polynomial $\mathcal{E}(G,S,q)$ has degree at most $\ell$ if and only if, for every $1\leq k\leq s$,

Let $G$ be a graph. Write $\underline{f}(G)$ for the minimum fort size of $G$, let $\mathrm{MObs}(G;i)$ denote the marginal observance at step $i$, and let $\gamma_P(G)$ denote the…

Let $K_{s,t}$ be the complete bipartite graph with part sizes $s$ and $t$, and let $\mathscr{PD}^{TAR}(K_{s,t})$ denote its power domination reconfiguration graph under token addit…

Let $G$ be any graph, and let $M(G)$ denote its middle graph. Write $\gamma_p(M(G))$ for the power domination number of $M(G)$ and $\gamma^{'}(G)$ for the edge domination number of…

Let $\mathcal{H}$ be a connected $r$-uniform hypergraph on $n$ vertices, and let $\gamma_p^k(\mathcal{H})$ denote its $k$-power domination number. Assume $k+r\leq n$. Chang–Roussel…

Quarter-order bound.

Let $l\ge 1$ and $k\ge l$ be integers, and let $G$ be a connected claw-free $(k+l+1)$-regular graph of order $n$. Let $\gamma_{p,k}(G)$ denote the minimum cardinality of a $k$-powe…

Let $G$ be a connected $r$-regular graph of order $n$, with $r\ge 3$, and let $k\ge 1$. Write $\gamma_{p,k}(G)$ for the minimum cardinality of a $k$-power dominating set of $G$. As…

Let $G$ be a graph of order $n$, and let $\overline{G}$ denote its complement. Assume that every component of both $G$ and $\overline{G}$ has order at least $3$. Write…