1. The signed (total) Roman domination problem on some classes of planar graphs— Convex polytopes.
- Author
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Zec, Tatjana, Matić, Dragan, and Djukanović, Marko
- Subjects
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PLANAR graphs , *RESEARCH personnel , *DOMINATING set , *POLYTOPES , *NEIGHBORS , *DEFINITIONS - Abstract
In last two decades, the basic Roman domination problem and the variants thereof have attracted many researchers from different fields to study these problems in both ways, theoretically and computationally. Function f : V → { − 1 , 1 , 2 } is the signed Roman domination function (SRDF) if and only if it satisfies the following conditions: (i) for each vertex v ∈ V of graph G the sum of the values assigned to v and all its neighbors is at least 1 and (ii) each vertex v ∈ V for which f (v) = − 1 , must be adjacent to at least one vertex u such that f (u) = 2. The minimum weight ∑ v ∈ V f (v) of all SRDFs on graph G is defined as signed Roman domination number ( γ sR (G)). Definition of the signed total Roman domination function (STRDF) follows the definition of SRDF with a minor change where condition (i) considers the sum of all values assigned to all neighbors of v to be at least one, for each vertex v. The minimum weight ∑ v ∈ V f (v) of all STRDFs on graph G is called signed total Roman domination number ( γ stR (G)). In this paper, we deal with the calculation of the signed (total) Roman domination numbers on a few classes of planar graphs from the literature. We give proofs for the exact values of the numbers γ sR (A n) and γ sR (R n) as well as the numbers γ stR (S n) and γ stR (T n). For some other classes of planar graphs, such as Q n , and T n ″ , lower and upper bounds on γ sR are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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