Double Roman Domination Number of Middle Graph

Abstract

For any graph G(V, E), a function f : V (G) 0, 1, 2, 3 is called Double Roman dominating function (DRDF) if the following properties holds, If f (v) = 0, then there exist two vertices v1, v2 ∈ N (v) for which f (v1) = f (v2) = 2 or there exist one vertex u ∈ N (v) for which f (u) = 3.∈ If f (v) = 1, then there exist one vertex u N (v) for which f (u) = 2 or Σ f (u) = 3. The weight of DRDF is the value w(f ) = v∈V (G) f (v). The minimum weight among all double Roman dominating function is called double Roman domination number and is denoted by γdR(G). In this article we initiated research on double Roman domination number for middle graphs. We established lower and upper bounds and also we characterize the double Roman domination number of middle graphs. Later we calculated numerical value of double Roman domination number of middle graph of path, cycle, star, double star and friendship graphs.

k-STRONG DEFENSIVE ALLIANCES IN GRAPHS

“…A set S ⊆ V is said to be a dominating set if ∪ s∈S N [s] = V . Some recent results on domination are found in [5,11]. A k-strong defensive alliance S in a graph is said to be a k-strong global defensive alliance if S is also a dominating set.…”