The Perfect Locating Signed Roman Domination of some Graphs
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Authors: A. D. AKWU, T. C. ADEFOKUN AND O. OYEWUMI
DOI: 10.46793/KgJMat2602.175A
Abstract:
In this paper, we introduce the concept of Perfect locating signed Roman dominating functions in graphs. A perfect locating signed Roman dominating PLSRD function of a graph G = (V,E) is a function f : V (G) →{−1, 1, 2} satisfying the conditions that for (i) every vertex v with f(v) = −1 is adjacent to exactly one vertex u with f(u) = 2; (ii) any pair of distinct vertices v,w with f(v) = f(w) = −1 does not have a common neighbor u with f(u) = 2 and (iii) f(v) + ∑ u∈N(v)f(u) ≥ 1 for any vertex v. The weight of PLSRD- function is the sum of its function values over all the vertices. The perfect locating signed Roman domination number of G denoted by γLSRP(G) is the minimum weight of a PLSRD- function in G. We present the upper and lower bonds of PLSRD- function for trees. In addition, for grid graph G, we show that γLSRP(G) ≤
|G|.
Keywords:
Perfect Roman domination, locating Roman domination, signed Roman domination, Cartesian product graph.
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