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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Kim, Kijung
Pusan National University
Electrical & Electronics Engineering

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The Italian bondage and reinforcement numbers of digraphs Kim, Kijung
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 13, No 2 (2025): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2025.13.2.9

Abstract

An Italian dominating function on a digraph D with vertex set V(D) is defined as a function f : V(D) → {0, 1, 2} such that every vertex v ∈ V(D) with f(v) = 0 has at least two in-neighbors assigned 1 under f or one in-neighbor w with f(w) = 2. The weight of an Italian dominating function f is the value ω(f) = f(V(D)) = ∑u∈V(D) f(u). The Italian domination number of a digraph D, denoted by γI(D), is the minimum taken over the weights of all Italian dominating functions on D. The Italian bondage number of a digraph D, denoted by bI(D), is the minimum number of arcs of A(D) whose removal in D results in a digraph D′ with γI(D′) > γI(D). The Italian reinforcement number of a digraph D, denoted by rI(D), is the minimum number of extra arcs whose addition to D results in a digraph D′ with γI(D′) < γI(D). In this paper, we initiate the study of Italian bondage and reinforcement numbers in digraphs and present some bounds for bI(D) and rI(D). We also determine the Italian bondage and reinforcement numbers of some classes of digraphs.
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