A two‐terminal graph is an undirected graph with two specified target vertices. If each nontarget vertex of a two‐terminal graph fails independently with the same fixed probability (and edges and target vertices are perfectly reliable), the two‐terminal node reliability is the probability that the target vertices are in the same connected component in the induced subgraph of all operational nodes. A two‐terminal graph is uniformly most reliable if its node reliability polynomial is greater than or equal to that of all other two‐terminal graphs with the same fixed number of vertices, n, and edges, m. In this paper, we show that there is always a uniformly most reliable two‐terminal graph. Furthermore, with the additional restriction that the distance between the target vertices is at least three, we completely classify which values of n and m produce a uniformly most reliable graph and which do not.

中文翻译：

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具有节点故障的最可靠的两端图

两端图是具有两个指定目标顶点的无向图。如果两端图的每个非目标顶点都以相同的固定概率独立失效（并且边和目标顶点是完全可靠的），则两端节点的可靠性是目标顶点在诱导的相同连接分量中的概率所有操作节点的子图。如果两端可靠性图的节点可靠性多项式大于或等于所有其他具有相同固定顶点数n和边线m的两端图，则两端图始终是最可靠的。。在本文中，我们表明始终有一个统一的最可靠的两端图。此外，在目标顶点之间的距离至少为3的附加限制下，我们可以完全分类n和m的哪些值生成一致最可靠的图，而哪些不生成。