25 problems

Let $SQ$ denote the infinite square grid, and let ERR:LD$(SQ)$ denote its error-correcting locating-dominating set density. Square-grid density conjecture. … The authors s…

Let $G$ be a finite undirected graph on $n$ vertices. A dominating set is a set of vertices such that every vertex is either in the set or adjacent to a vertex in the set. For…

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.

Total domination conjecture.

Matheson–Tarjan 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 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…

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 $T$ 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…

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…

Garijo et al.'s conjecture. Every twin-free graph $G$ of order $n$ without isolated vertices satisfies

Existence conjecture. Let $H$ be a finite path or a Cartesian product of two finite paths. Then a $t$-PDDS$[H]$ in $\Lambda_n$ exists if and only if at least one of the following h…

King and Pelsmajer's conjecture. For any constant $t$, there exists $n_t$ such that every $n$-vertex plane triangulation with $n>n_t$ and at most $t$ vertices of degree other than…

Let $\Delta$ be the triangular lattice. A $K_2$-QPDS $S$ in $\Delta$ has an associated hexagonal structure ${\mathcal P}(S)$ whose hexagons have types $1$, $2$, and $3$. Mortar-and…

A triangulation is a plane graph in which every face, including the outer face, is bounded by a triangle. The degree of a vertex is its number of incident edges. The bounded-except…

A triangulation is a plane graph in which every face, including the outer face, is bounded by a triangle. A dominating set of a graph is a set containing every vertex or a neighbor…