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k-Domination ivariants on Kneser graphs
Boštjan Brešar, Tanja Dravec, María Gracia Cornet, Michael A. Henning, 2025, original scientific article

Abstract: In this follow-up to work of M.G. Cornet and P. Torres from 2023, where the k-tuple domination number and the 2-packing number in Kneser graphs K(n, r) were studied, we are concerned with two variations, the k-domination number, γ_k(K(n, r)), and the k-tuple total domination number, γ_{t × k}(K(n, r)), of K(n, r). For both invariants we prove monotonicity results by showing that γ_k(K(n, r)) ≥ γ_k(K(n + 1, r)) holds for any n ≥ 2(k + r), and γ_{t × k}(K(n, r)) ≥ γ_{t × k}(K(n + 1, r)) holds for any n ≥ 2r + 1. We prove that γ_k(K(n, r)) = γ_{t × k}(K(n, r)) = k + r when n ≥ r(k + r), and that in this case every γ_(k)-set and γ_(t × k)-set is a clique, while γ_k(r(k + r) − 1, r) = γ_{t × k}(r(k + r) − 1, r) = k + r + 1, for any k ≥ 2. Concerning the 2-packing number, ρ₂(K(n, r)), of K(n, r), we prove the exact values of ρ₂(K(3r − 3, r)) when r ≥ 10, and give sufficient conditions for ρ₂(K(n, r)) to be equal to some small values by imposing bounds on r with respect to n. We also prove a version of monotonicity for the 2-packing number of Kneser graphs.
Keywords: Kneser graphs, k-domination, k-tuple total domination, 2-packing
Published in RUP: 22.10.2025; Views: 746; Downloads: 8
Full text (375,34 KB)

Indicated total domination game
Michael A. Henning, Douglas F. Rall, 2025, original scientific article

Abstract: A vertex u in a graph G totally dominates a vertex v if u is adjacent to v in G. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The indicated total domination game is played on a graph G by two players, Dominator and Staller, who take turns making a move. In each of his moves, Dominator indicates a vertex v of the graph that has not been totally dominated in the previous moves, and Staller chooses (or selects) any vertex adjacent to v that has not yet been played, and adds it to a set D that is being built during the game. The game ends when every vertex is totally dominated, that is, when D is a total dominating set of G. The goal of Dominator is to minimize the size of D, while Staller wants just the opposite. Providing that both players are playing optimally with respect to their goals, the size of the resulting set D is the indicated total domination number of G, denoted by γti(G). In this paper we present several results on indicated total domination game. Among other results we prove that the indicated total domination number of a graph is bounded below by the well studied upper total domination number.
Keywords: total domination game, indicated total domination game
Published in RUP: 21.10.2025; Views: 924; Downloads: 8
Full text (378,61 KB)

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