AI 中文总结
本文确定了使连通无导出-H图的每个顶顶点均为Gallai顶点的图H，结论为H是阶数至多4的线性森林或P₃+2P₁。
AI 中文摘要
图中的顶点若在图中具有最大度，则称为顶顶点；若属于图的每条最长路径，则称为Gallai顶点。Golan和Shan证明，任何连通无导出-2P₂图的每个顶顶点都是Gallai顶点。Long、Milans和Munaro随后指出，若每个连通无导出-H图都有一个Gallai顶点，则H必为阶数至多9的线性森林；他们还证明，对于每个阶数至多4的线性森林H，任何连通无导出-H图的每个顶顶点都是Gallai顶点。本文确定了所有满足“任何连通无导出-H图的每个顶顶点都是Gallai顶点”的图H，结果表明，当且仅当H是阶数至多4的线性森林或H=P₃+2P₁时，该结论成立。
英文摘要
A vertex of a graph is called a top vertex if it has maximum degree in the graph. A vertex of a graph is called a Gallai vertex if it belongs to every longest path of the graph. Golan and Shan proved that every top vertex of any connected induced-$2P_2$ free graph is a Gallai vertex. Long, Milans, and Munaro subsequently showed that if every connected induced-$H$ free graph has a Gallai vertex, then $H$ must be a linear forest of order at most nine. They also proved that, for every linear forest $H$ of order at most four, every top vertex of any connected induced-$H$ free graph is a Gallai vertex. In this paper, we determine the graphs $H$ for which every top vertex of any connected induced-$H$ free graph is a Gallai vertex. Our result shows that this holds precisely when $H$ is a linear forest of order at most four or $H=P_3+2P_1$.
Comments17 pages; 5 figures
