A Note on Total and Paired Domination of Cartesian Product Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in $G$. Vizing’s conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G) \gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G) \gamma(H) \leq 2\gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

💡 Research Summary

The paper investigates domination parameters—ordinary domination (γ), total domination (γ_t), and paired domination (γ_pr)—in Cartesian product graphs. Starting from Vizing’s conjecture (γ(G)·γ(H) ≤ γ(G□H)) and the Clark‑Suen bound (γ(G)·γ(H) ≤ 2·γ(G□H)), the authors modify the double‑counting technique to obtain analogous inequalities for the other two domination concepts and to extend all results to n‑fold Cartesian products.

The authors first introduce the necessary notation. For a set S⊆V(G□H) they define projections Φ_G(S) and Φ_H(S) onto the factor graphs. They also define G‑neighbourhoods and H‑neighbourhoods of a vertex (g,h) in the product, and partition the edge set of G□H into G‑edges and H‑edges. For the n‑fold product A₁□…□A_n they similarly define i‑neighbourhoods and a partition of edges into n families E_i.

Two elementary matrix propositions are proved. Proposition 1 states that any binary matrix either has a 1 in every column or a 0 in every row. Proposition 2 generalizes this to d₁×…×d_n n‑ary matrices, introducing the notion of a j‑matrix (a matrix where each (d₁×…×d_{j‑1}×1×d_{j+1}×…×d_n) sub‑matrix contains the entry j). These propositions are the combinatorial backbone of the later arguments.

Theorem 1 (mixed domination): For graphs without isolated vertices, \