More Reliable Graphs Are Not Always Stronger

A graph G$$ G $$ is stronger than a graph H$$ H $$ if G$$ G $$ has at least as many connected spanning subgraphs of size k$$ k $$ as H$$ H $$ for any positive integer k$$ k $$. Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph G$$ G $$ is uniformly more reliable than H$$ H $$ if its reliability polynomial is greater than or equal to the reliability polynomial of H$$ H $$ for all probabilities. As a direct consequence of the definition, a sufficient condition for G$$ G $$ to be uniformly more reliable than H$$ H $$ is for G$$ G $$ to be stronger than H$$ H $$. In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, Gk$$ {G}_k $$ and Hk$$ {H}_k $$, such that Gk$$ {G}_k $$ is uniformly more reliable than Hk$$ {H}_k $$ but is not stronger than Hk$$ {H}_k $$.