Cost-Effective Network Optimization using Middle Weak Roman Domination with Python Integration

Middle weak Roman domination is an important graph theory concept that has important applications in defense, network design, optimization, and resource management. A middle weak Roman dominating function assigns to every vertex of a graph weights from the set {0,1,2,3} under certain conditions: zero-weight vertices must be neighbors of at least one positive-weighted vertex; one-weight vertices may have at most one zero-weight neighbor; two-weight vertices may have at most two zero-weight neighbors; and three-weight vertices may have any number of zero-weight neighbors. The total weight of assigned weights determines the weight of the function, and the least possible weight among all valid functions is referred to as the middle weak Roman domination number. This paper explores the calculation of this number across various graph classes, including complete graphs, paths, cycles, wheels, stars, sun graphs, complete binary trees, grids, and lollipop graphs. Python pseudo-code is provided for efficient computation.