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A graph is well-dominated if all its minimal dominating sets have the same cardinality. In this paper we prove that at least one factor of every connected, well-dominated Cartesian product is a complete graph, which then allows us to give a complete characterization of the connected, well-dominated Cartesian products if both factors have order at least $2$. In particular, we show that $G\,\Box\,H$ is well-dominated if and only if $G\,\Box\,H = P_3 \,\Box\,K_3$ or $G\,\Box\,H= K_n \,\Box\,K_n$ for some $n\ge 2$., 17 pages

Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$ using this process is known as a power dominating set of $G$, and the power domination number of $G$, $\gamma_P(G)$, is the minimum number of vertices in a power dominating set. We introduce a new partition on the vertices of a graph to provide a lower bound for the power domination number. We also consider the power domination number of the Cartesian product of two graphs, $G \Box H$, and show certain graphs satisfy a Vizing-like bound with regards to the power domination number. In particular, we prove that for any two trees $T_1$ and $T_2$, $\gamma_P(T_1)\gamma_P(T_2) \leq \gamma_P(T_1 \Box T_2)$.

The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $\gamma_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $\gamma_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $\gamma_P(G) \le n/6$, and this bound is sharp. We also provide new bounds on $\gamma_P(G \Box H)$ where $G\Box H$ is the Cartesian product of graphs $G$ and $H$. In the specific case that $G$ and $H$ are trees whose power domination number and domination number are equal, we show the Vizing-like inequality holds and $\gamma_P(G \Box H) \ge \gamma_P(G)\gamma_P(H)$.

Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw-free cubic graph $G$, $Z(G) \le \frac{1}{3}|V(G)| + 1$. We show that if $G$ is $2$-edge-connected, claw-free, and cubic, then $Z(G) \le \left\lceil\frac{5n(G)}{18}\right\rceil+1$. We also study a similar graph invariant known as the loop zero forcing number of a graph $G$ which happens to be the dual invariant to the Grundy domination number of $G$. Specifically, we study the loop zero forcing number in two particular types of planar graphs.

A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $\gamma(D)$ (respectively,$\gamma_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $\rho(D)$ (respectively,$\rho^{\rm o}(D)$). We prove that in digraphs whose underlying graphs have girth at least $7$, the closed (respectively,open) in-neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree $T$ (that is, a digraph whose underlying graph is a tree), $\gamma_t(T)=\rho^{\rm o}(T)$ and $\gamma(T)=\rho(T)$. By using the former equality we then prove that $\gamma_t(G\times T)=\gamma_t(G)\gamma_t(T)$, where $G$ is any digraph and $T$ is any ditree, each without a source vertex, and $G\times T$ is their direct product. From the equality $\gamma(T)=\rho(T)$ we derive the bound $\gamma(G\mathbin{\Box} T)\ge\gamma(G)\gamma(T)$, where $G$ is an arbitrary digraph, $T$ an arbitrary ditree and $G\mathbin{\Box} T$ is their Cartesian product. In general digraphs this Vizing-type bound fails, yet we prove that for any digraphs $G$ and $H$, where $\gamma(G)\ge\gamma(H)$, we have $\gamma(G \mathbin{\Box} H) \ge \frac{1}{2}\gamma(G)(\gamma(H) + 1)$. This inequality is sharp as demonstrated by an infinite family of examples. Ditrees $T$ and digraphs $H$ enjoying $\gamma(T\mathbin{\Box} H)=\gamma(T)\gamma(H)$ are also investigated., Comment: 22 pages and 5 figures

A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from $P_2$. We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least $6$. Finally, we show that the recognition of the graphs with a unique maximum open packing is polynomially equivalent to the recognition of the graphs with a unique maximum independent set, and we prove that the complexity of both problems is not polynomial, unless P=NP., 16 pages, 4 figures