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Efficient total domination and related invariants in torus graphs.

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Τίτλος: Efficient total domination and related invariants in torus graphs.
Συγγραφείς: Wehrmann, Moritz1 (AUTHOR) wehrmann@math2.rwth-aachen.de, Koster, Arie M.C.A.1 (AUTHOR) koster@math2.rwth-aachen.de
Πηγή: Discrete Applied Mathematics. Aug2026, Vol. 389, p106-118. 13p.
Θεματικοί όροι: *Dominating set, *Regular graphs, *Mixed integer linear programming
Περίληψη: For a graph G , a set of vertices S is an efficient total k -dominating set (ET k D set) if every vertex of G is adjacent to exactly k vertices in S. A torus graph is the Cartesian product C m □ C n of two cycles. In this work, we characterize all torus graphs with an ET k D set for k = 1 , 2 , 3. In particular, we prove following conjecture of Kuziak, Peterin, and Yero (2014): C m □ C n has an ET1D set if and only if m ≡ n ≡ 0 (mod 4). This allows us to determine the total domatic number and the injective chromatic number of all torus graphs. Additionally, we characterize the signed total Roman domatic number of 4-regular graphs in terms of ET1D and ET2D sets, and consequently obtain exact values for all torus graphs. • Characterization of torus graphs with an efficient total dominating set. • Pattern-based approach to coloring torus graphs via integer linear programming. • Signed total Roman domatic numbers of quartic graphs via efficient total domination. [ABSTRACT FROM AUTHOR]
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  Data: Efficient total domination and related invariants in torus graphs.
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  Data: <searchLink fieldCode="JN" term="%22Discrete+Applied+Mathematics%22">Discrete Applied Mathematics</searchLink>. Aug2026, Vol. 389, p106-118. 13p.
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  Data: For a graph G , a set of vertices S is an efficient total k -dominating set (ET k D set) if every vertex of G is adjacent to exactly k vertices in S. A torus graph is the Cartesian product C m □ C n of two cycles. In this work, we characterize all torus graphs with an ET k D set for k = 1 , 2 , 3. In particular, we prove following conjecture of Kuziak, Peterin, and Yero (2014): C m □ C n has an ET1D set if and only if m ≡ n ≡ 0 (mod 4). This allows us to determine the total domatic number and the injective chromatic number of all torus graphs. Additionally, we characterize the signed total Roman domatic number of 4-regular graphs in terms of ET1D and ET2D sets, and consequently obtain exact values for all torus graphs. • Characterization of torus graphs with an efficient total dominating set. • Pattern-based approach to coloring torus graphs via integer linear programming. • Signed total Roman domatic numbers of quartic graphs via efficient total domination. [ABSTRACT FROM AUTHOR]
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