Partitioning the vertices of a cubic graph into two total dominating sets

Disjoint Paired-Dominating sets in Cubic Graphs

“…In this note, we strengthen the result of [6] from another point of view by proving the following theorem.…”

“…Let us note that Theorem 1 is not a kind of strengthening of the result quoted from [6] that would follow by changing a few words in earlier arguments; in fact it needs a substantially different approach. Namely, the proof in [6] strongly uses hypergraph theorems, while our proof is purely graph-theoretic, with a more specific analysis of the structure.…”

“…Very recently, Desormeaux et al [6] proved that every claw-free cubic graph has two disjoint total dominating sets, while Akbari et al [1], independently and simultaneously, proved the same property for a slightly wider subclass of cubic graphs. In this note, we strengthen the result of [6] from another point of view by proving the following theorem.…”

“…Namely, the proof in [6] strongly uses hypergraph theorems, while our proof is purely graph-theoretic, with a more specific analysis of the structure. In particular, we first proceed by induction, then define an auxiliary graph and apply Vizing’s theorem to finish the proof.…”

Maker–Breaker total domination game

“…We observe that every tree G satisfies tdom(G) = 1, and so by Theorem 3.1, the S-game is won by Staller in the class of trees. As observed in [9], there are infinitely many examples of connected cubic graphs G with tdom(G) = 1. The Heawood graph, G 14 , shown in Figure ??…”