Total domination–annihilation conjecture for graphs with pendant vertices

Prove or refute that every connected nontrivial graph G satisfies the inequality γ_t(G) ≤ a(G) + 1, with particular attention to the remaining case of graphs having minimum degree one and therefore containing pendant vertices.

Background

The paper studies the conjecture that the total domination number γ_t(G) is at most one more than the annihilation number a(G) for every connected nontrivial graph. It proves the conjecture for all connected graphs with minimum degree two, while the case of minimum degree at least three was already known. Consequently, any unresolved counterexample must have minimum degree one and contain a pendant vertex.

The paper notes that existing structural restrictions on minimum counterexamples may be useful for resolving this final case, but does not determine whether such counterexamples exist. Thus, the global conjecture remains unresolved only for connected graphs containing pendant vertices.

References

The following conjecture was posed in a slightly different form by Graffiti.pc and was later reformulated by Desormeaux, Haynes, and Henning. If $G$ is a connected nontrivial graph, then $$\gamma_t(G)\le a(G)+1.$$

— Settling the total domination-annihilation conjecture for graphs with minimum degree two  (2609.10795 - Jakovac, 9 Sep 2026) in Section 1, Introduction; Section 4, “Sharpness and possible counterexamples”