Abstract
A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.
Highlights
Graphs considered in this paper are all simple, finite and undirected.We consider a graph G = (V, E) having no isolated vertices
We find the exact value of the binary locating-dominating number for this family of convex polytope
We find the exact value of the the binary locating-dominating number of Hn0
Summary
Graphs considered in this paper are all simple, finite and undirected. We consider a graph G = (V, E) having no isolated vertices. In this terminology, D is called dominating set if the sum of weights of closed neighborhoods of any vertex in G is at least one. Charon et al [3] studied the minimum cardinalities of r-locating-dominating and r-identifying codes for cycles and chains. They characterized the extremal values for these parameters. Imran et al [17,18,19] studied the problem of minimum metric dimension for different infinite families of convex polytopes. Considered minimal double resolving sets and the strong metric dimension problem for some families of convex polytopes. Simić et al [23] studied the problem of binary locating-dominating number of some convex polytopes. Other graph-theoretic parameters having potential applications in chemistry are studied in [24,25,26,27]
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