On mixed double Roman domination number and ۲-independence number of graphs

Let G=(V(G),E(G)) be a simple graph. A mixed double Roman dominating function (MDRDF) on G is a function f:V(G)∪E(G) → {۰,۱,۲,۳} such that (i) if f(u)=۰, then u has at least two neighbors assigned ۲ under f or one neighbor w with f(w)=۳; (ii) if f(u)=۱, then u must have at least one neighbor w with f(w)≥ ۲. The weight of f is equal to Σu∈V(G)∪E(G)f(u). The mixed double Roman domination number γdR*(G) is the minimum weight among all MDRDF's of G. A subset X of V(G) is a ۲-independent set of G if the subgraph induced by the vertices of X is isomorphic to rK۲∪sK۱, where ۲r+s=|X|. The maximum cardinality of a ۲-independent set of G is the ۲-independence number β۲(G). In this paper, it is shown that these parameters are incomparable, in general. Also, we prove that either T is a path or a tree with a quasi-diagonal path u۱,…, uk such that d(u۲)≠ ۲ or d(u۳)≠ ۲, then γdR*(T)≤ ۲β۲(T). Moreover, all extremal trees in this family that attaining equality are characterized.

کلیدواژه ها:

۲-independence number ، mixed double Roman domination number ، tree