2025/10/01 by Jing Tian, Tian, Jing, Pakanun Dokyeesun +3
Engineering · Computer Science · #Aerospace Engineering and Control Systems #Optimization and Variational Analysis #Robotic Path Planning Algorithms
paper · pdf · doi:10.48550/arxiv.2510.01294
Let \rm gp\rm t(G), \rm gp\rm o(G), and \rm gp\rm d(G) be the total, the outer, and the dual general position number of a graph G, respectively. This paper investigates how removing a vertex or removing an edge affects these graph invariants. It is proved that if x is not a cut vertex, then \rm gp\rm t(G) -1 ≤ \rm gp\rm t(G-x) ≤ \rm gp\rm t(G) + \rm degG(x). On the other hand, \rm gp\rm o(G-x) and \rm gp\rm d(G-x) can be respectively arbitrarily larger/smaller than \rm gp\rm o(G) and \rm gp\rm d(G). On the positive side, it is proved that if x lies in some \rm gp\rm o-set, then \rm gp\rm o(G)-1 ≤ \rm gp\rm o(G-x), and that if x is not a cut vertex and lies in some \rm gp\rm d-set of G, then \rm gp\rm d(G)-1 ≤ \rm gp\rm d(G-x). For the edge removal, it is proved that (i) \rm gp\rm t(G) -|S(G)e| ≤ \rm gp\rm t(G-e) ≤ \rm gp\rm t(G) +2, where S(G)e is the set of simplicial vertices adjacent to both endvertices of e, (ii) \rm gp\rm o(G)/2≤ \rm gp\rm o(G-e)≤ 2\rm gp\rm o(G), and (iii) that \rm gp\rm d(G) - \rm gp\rm d(G-e) can be arbitrarily large. All bounds are demonstrated to be sharp.