Jillian Cervantes, “Optimal Resource Placement: Dominating Sets for a Family of Semiregular Tiling Graphs”
Mentor: Pamela Harris, Mathematical Sciences
Poster #220
Let G = (V, E) be a simply connected undirected finite graph with vertex set V and edge set E. For a positive integer k, a set of vertices D ⊆ V is said to be a distance -dominating set if for all v ∈ V, there exists u ∈ D such that d (u, v) ≤ k . We let γk (G) denote the smallest cardinality among all k -dominating sets of G. We consider a tessellation of the plane using regular octagons and squares, and create a graph for each m, n, ∈ Z≥0 consisting of m rows and n columns of regular octagons. We denote these graphs by Hm,n and establish formulas to enumerate the vertices in Hm,n and give upper and lower bounds for γ1 (Hm,n). We also extend our work to the setting of (t, r) broadcast domination and provide some related results.