2024/06/25 by Changchang Dong, Dong, Changchang, Yubao Guo +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2406.17368
openalex publication_date 2024/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a finite simple digraph D with vertex set V(D). An Italian dominating function (IDF) on D is a function f:V(D)→\0,1,2\ satisfying every vertex u with f(u)=0 has an in-neighbor v with f(v)=2 or two in-neighbors w and z with f(w)=f(z)=1. A total Italian dominating function (TIDF) on D is an IDF f such that the subdigraph D[\ u | f(u)≥ 1\] contains no isolated vertices. The weight ω(f) of a TIDF f on D is ∑u∈ V(D)f(u). The total Italian domination number of D is γtI(D)=min\ ω(f) | f is a TIDF on D\. In this paper, we present bounds on γtI(D), and investigate the relationship between several different domination parameters. In particular, we give the total Italian domination number of the Cartesian products P2\Box Pn and P3\Box Pn, where Pn represents a dipath with n vertices.