Revisiting the outer-weakly convex domination number in graph products
Bijo S. Anand, Ullas Chandran S. V., Jonecis A. Dayap, Leomarich F. Casinillo, Karen Luz P. Yap
TL;DR
This work investigates the outer-weakly convex domination number $\widetilde{ \gamma}_{wcon}(G)$, a convexity-based domination parameter, across three standard graph products: Cartesian, strong, and lexicographic. The authors establish projection-based properties that yield lower and upper bounds for each product, and in several cases prove sharp results or exact values (e.g., $\widetilde{ \gamma}_{wcon}(G \boxtimes H)$ bounds and $\widetilde{ \gamma}_{wcon}(G \circ H)$ inequalities with the $\mathcal{P}_G$ parameter). Key contributions include tight bounds for $G \Box H$ and $G \boxtimes H$, plus a constructive upper bound for $G \circ H$ that tightens when $\mathcal{P}_G=0$. These results enhance understanding of how outer-weak convexity interacts with product operations, guiding future work on restricted graph classes and exact determinations. The findings have implications for convexity-driven domination in network design and analysis where graph products model composite systems.
Abstract
Let $G = (V, E)$ be a simple undirected graph. A set $C \subseteq V(G)$ is weakly convex of graph $G$ if for every two vertices $u,v\in G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an outer-weakly convex dominating set if it is dominating set and every vertex not in $C$ is adjacent to some vertex in $C$ and a set $V(G)\setminus C$ is weakly convex. The outer-weakly convex domination number of graph $G$, denoted by $\widetilde{ γ}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating vertex set of graph $G$. In this paper, we determined the outer-weakly convex domination number of two graphs under the cartesian, strong and lexicographic products, and discuss some important combinatorial findings.