1
Department of Mathematics, Za. C., IAU, Zanjan, Iran.
2
Department of Mathematics,Payame Noor University,Tehran,Iran
Abstract
The efficient placement of control points (sensors, monitoring stations, valves, or traffic controllers) in urban infrastructure networks is a fundamental problem in civil engineering. This paper introduces the graph-theoretic concept of the hub number as a mathematical tool for modeling and optimizing such placements. A hub set in a graph is a vertex subset such that every pair of vertices outside the set is connected by a path whose internal vertices all lie in the set; the hub number h(G) is the minimum cardinality of such a set. We focus on grid graphs G_{m,n\ }=P_m □ P_n, which serve as idealized models of regular urban street networks and pipeline systems. Drawing on established results for the hub numbers of Cartesian products of paths, we present exact values and upper bounds for several families of grid graphs and interpret them in engineering terms. A hypothetical case study on rectangular grid networks demonstrates how the hub number provides lower bounds on the number of control points required to guarantee network-wide path-mediated connectivity. Comparisons with classical domination parameters highlight the usefulness of the hub-set approach. The results offer civil engineers a rigorous criterion for the preliminary design of monitoring and control systems in grid-like urban infrastructures.
Naserian,O and Eini,M . (2026). Application of Hub Number in Optimal Design of Urban Infrastructure Networks: A Graph-Theoretic Approach for Control Point Placement. (e250727). Civil and Project journal, (), e250727 doi: 10.22034/cpj.2026.596889.1466
MLA
Naserian,O , and Eini,M . "Application of Hub Number in Optimal Design of Urban Infrastructure Networks: A Graph-Theoretic Approach for Control Point Placement" .e250727 , Civil and Project journal, , , 2026, e250727. doi: 10.22034/cpj.2026.596889.1466
HARVARD
Naserian O, Eini M. (2026). 'Application of Hub Number in Optimal Design of Urban Infrastructure Networks: A Graph-Theoretic Approach for Control Point Placement', Civil and Project journal, (), e250727. doi: 10.22034/cpj.2026.596889.1466
CHICAGO
O Naserian and M Eini, "Application of Hub Number in Optimal Design of Urban Infrastructure Networks: A Graph-Theoretic Approach for Control Point Placement," Civil and Project journal, (2026): e250727, doi: 10.22034/cpj.2026.596889.1466
VANCOUVER
Naserian O, Eini M. Application of Hub Number in Optimal Design of Urban Infrastructure Networks: A Graph-Theoretic Approach for Control Point Placement. CPJ. 2026;():e250727 (In Persian). doi: 10.22034/cpj.2026.596889.1466