A subset D■V(G)in a graph G is a dominating set if every vertex in V(G)I D is adjacent to at least one vertex of S.A subset S■V(G)in a graph G is a 2-independent set if△(G[S])<2.The 2-independence numberα2(G)is the order of a largest 2-independent set in G.Further,a subset D■V(G)in a graph G is a 2-independent dominating set if D is both dominating and 2-independent.The 2-independent domination number i^(2)(G)is the order of a smallest 2-independent dominating set in G.In this paper,we characterize all trees T of order n with i^(2)(T)=n/2.Moreover,we prove that for any tree T of order n≥2,i^(2)(T)≤2/3α2(T),and this bound is sharp.
Soft actuators have garnered substantial attention in current years in view of their potential appliances in diverse domains like robotics,biomedical devices,and biomimetic systems.These actuators mimic the natural movements of living organisms,aiming to attain enhanced flexibility,adaptability,and versatility.On the other hand,angle-independent structural color has been achieved through innovative design strategies and engineering approaches.By carefully controlling the size,shape,and arrangement of nanostructures,researchers have been able to create materials exhibiting consistent colors regardless of the viewing angle.One promising class of materials that holds great potential for bioinspired soft actuators is MXenes in view of their exceptional mechanical,electrical,and optical properties.The integration of MXenes for bioinspired soft actuators with angle-independent structural color offers exciting possibilities.Overcoming material compatibility issues,improving color reproducibility,scalability,durability,power supply efficiency,and cost-effectiveness will play vital roles in advancing these technologies.This perspective appraises the development of bioinspired MXene-centered soft actuators with angleindependent structural color in soft robotics.
