AI 中文总结
本文证明了对最小度为二的连通图，全控制数不超过湮灭数加一，从而解决了该猜想在最小度为二情形下的遗留问题。
AI 中文摘要
图$G$的全控制数$\gamma_t(G)$是满足$G$中每个顶点在$D$中都有一个邻居的集合$D\subseteq V(G)$的最小基数。湮灭数$a(G)$是使得$G$的$k$个最小度之和至多为$|E(G)|$的最大整数$k$。一个著名的猜想，源于此http URL，后来由Desormeaux、Haynes和Henning明确表述，断言对于每个连通非平凡图$G$，有$\gamma_t(G)\le a(G)+1$。该猜想已知对于最小度至少为三的图以及若干具有一度或二度顶点的图类成立。在本文中，我们解决了最小度为二的情形。更精确地，我们证明对于每个满足$\delta(G)=2$的连通图$G$，有$\gamma_t(G)\le a(G)+1$。证明结合了关于全控制数的两个尖锐界与对湮灭数的一个估计。此外，在某些特定情况下，更强的不等式$\gamma_t(G)\le a(G)$成立。
英文摘要
The total domination number $γ_t(G)$ of a graph $G$ is the minimum cardinality of a set $D\subseteq V(G)$ such that every vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ is the largest integer $k$ for which the sum of the $k$ smallest degrees of $G$ is at most $|E(G)|$. A well-known conjecture, originating from Graffiti.pc and later formulated explicitly by Desormeaux, Haynes, and Henning, asserts that $γ_t(G)\le a(G)+1$ for every connected nontrivial graph $G$. The conjecture is known for graphs of minimum degree at least three and for several classes of graphs having vertices of degree one or two. In this paper we settle the minimum-degree-two case. More precisely, we prove $γ_t(G)\le a(G)+1$ for every connected graph $G$ with $δ(G)=2$. The proof combines two sharp bounds on the total domination number with an estimate for the annihilation number. Moreover, in some specific cases, the stronger inequality $γ_t(G)\le a(G)$ holds.
