Outer independent signed double Roman domination

Suppose [ 3 ] = { 0 , 1 , 2 , 3 } and [ 3 - ] = { - 1 , 1 , 2 , 3 } . An outer independent signed double Roman dominating function (OISDRDF) of a graph Γ is function l : V ( Γ ) → [ 3 - ] for which (i) each vertex t with l ( t ) = - 1 is joined to at least two vertices labeled a 2 or to at least one vertex z with l ( z ) = 3 , (ii) each vertex t with l ( t ) = 1 is joined to at least a vertex z with l ( z ) ≥ 2 , (iii) l ( N [ t ] ) = ∑ w ∈ N [ t ] l ( w ) ≥ 1 occurs for each vertex t , (iv) the set of vertices labeled - 1 under l is an independent set. The weight of an OISDRDF is the sum of its function values over all vertices, and the outer independent signed double Roman domination number (OISDRD-number) γ sdR oi ( Γ ) is the minimum weight of an OISDRDF on Γ . We first show that determining the number γ sdR oi ( Γ ) is NP-complete for bipartite and chordal graphs. Then we provide exact values of this parameter for paths and cycles. Moreover, we show that for trees T of order n ≥ 3 , γ sdR oi ( Γ ) ≤ n - 1 , and we characterize extremal trees attaining this bound.