Signed Total Strong Roman Domination in Graphs

Abstract

Let G = ( V, E ) be a finite and simple graph of order n and maximum degree ∆ . A signed total strong Roman dominating function on G is a function f : V → {− 1 , 1 , 2 , . . . , (cid:100) ∆ / 2 (cid:101) + 1 } satisfying the conditions: (i) for every vertex v of G , (cid:80) u ∈ N ( v ) f ( u ) ≥ 1 , where N ( v ) is the open neighborhood of v , and (ii) every vertex v satisfying f ( v ) = − 1 is adjacent to at least one vertex u such that f ( u ) ≥ 1 + (cid:6) | N ( u ) ∩ V − 1 | / 2 (cid:7) , where V − 1 = { v ∈ V | f ( v ) = − 1 } . The signed total strong Roman domination number of G , γ tssR ( G ) , is the minimum weight of a signed total strong Roman dominating function. In this paper, some bounds for this parameter are presented.