34 problems

Total domination conjecture.

Ultra log-concavity conjecture. For all $n\geq 0$ and $\ell\geq 1$, both $(\gamma(P_n^\ell,k))_{k\geq 0}$ and $(\gamma(C_n^\ell,k))_{k\geq 0}$ are ultra log-concave.

Let $G$ be a graph with $n\geq 2$ vertices, and let $\operatorname{avd}(G)$ denote its average order of dominating sets. If $G$ has no isolated vertices, then Beaton–Brown's conjec…

Let $\Gamma(\mathbb{Z}_{2n+1}^n,\mathcal{U})$ be the Cartesian product of $n$ cycles of length $2n+1$, where $2n+1$ is prime, and let $\mathcal{F}$ be the family of all minimum dom…

Let $G$ be a graph on $n$ vertices. A dominating set is a subset $S$ of the vertices such that every vertex is either in $S$ or adjacent to a vertex in $S$, and let $\bar{G}$ denot…

Let $G$ be a cycle on $n\ge 3$ vertices. Its dominion is the invariant denoted by $\zeta(G)$. Cycle dominion conjecture. If $G$ is a cycle on $n\ge 3$ vertices, then … The cases…

Connected domination conjecture.

Let $\chi_n$ denote the parameter defined in the paper for the $n$-dimensional hypercube. Growth and limit conjecture. The sequence $\chi_n$ satisfies both of the following propert…

Dejter's four-operation conjecture. Every ETC of $\Gamma$ is obtained via the iterative genus-increasing procedure of Corollary, departing from $Q_3$, through four constructive ope…

Henning–Löwenstein–Rautenbach conjecture. Every connected subcubic graph $G$ except the three graphs $H_1$, $H_2$, and $H_3$ satisfies

Let $G$ be a $C_4$-free cubic graph, and let $\textrm{DET:OLD}\%(G)$ denote the minimum density of an error-detecting open-locating-dominating set in $G$. DET:OLD density conjectur…

Dominating-set reformulation. If

Let $G=(V,E)$) be the infinite king grid with vertex set $V=\mathbb{Z}\times\mathbb{Z}$, in which two vertices are adjacent exactly when their Euclidean distance is at most…

For a graph $G$, let $D(G,x)$ be its domination polynomial, and call the index of a largest coefficient its mode. A graph is star-like in the sense intended by the source. Star-lik…

For a graph $G$, let $D(G,x)$ denote its domination polynomial. A graph is star-like in the sense intended by the source, and $D(G,x)$ is real-rooted when all of its roots are real…

Let $T$ be a tree with domination number $gamma$, and let the number of its minimum dominating sets be measured as a function of $gamma$. Alvarado et al.'s conjecture. A tree with…

Twin-free graph bounds conjecture. The two bounds

For every positive integer $k$, let $f(k)$ be an integer such that every complete multidigraph whose arcs are the union of $k$ quasi-orders has a dominating set of size at most…

Let $G$ be a twin-free graph of order $n$, and let a locating-dominating set be a set of vertices that is both locating and dominating. Write $\gamma^{LD}(G)$ for the minimum size…

Let $D$ be a twin-free digraph of order $n$, and let $\gamma^{LD}(D)$ denote the minimum size of a locating-dominating set of $D$. Foucaud, Heydarshahi and Parreau proved…

Let $G$ be a non-empty graph, let $\operatorname{avd}(G)$ denote the average order of a dominating set of $G$, and let $G\setminus e$ be the graph obtained by deleting an edge $e$…

In-dominating-set conjecture. There exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that, for every $k\geq1$, if

Let $G$ be a nonempty graph. For a vertex $v$ and an edge $e$, write $G-v$ and $G-e$ for the graphs obtained by deleting $v$ and $e$, respectively. The monotonicity sandwich conjec…

Let $G$ be a graph, let $\alpha(G)$ denote its independence number, and let a swap set mean the graph structure defined in the paper that supports the relevant disjoint dominating…

Let $G$ be a connected graph containing a swap set. Let $DD_{\operatorname{m}}(G)$ denote the minimum size of a disjoint dominating set whose two parts are joined by a perfect matc…