AbstractThe stability of a (communication or transportation) network composed of (processing) nodes and (communication or transportation) links is of prime importance to network designers. As the network begins losing links or nodes, eventually, there is a loss in its effectiveness. Thus, it is desirable that networks be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconfiguration of the network after disruption. Many graph theoretical parameters have been used in the past to describe the stability of networks, including the vertex‐connectivity and edge‐connectivity, toughness and edge‐toughness, integrity and edge‐integrity, and tenacity. In this paper, we study the edge‐tenacity of graphs. We will be primarily interested in edge‐tenacious graphs, which can be considered very stable and are somewhat analogous in edge‐tenacity to honest graphs in edge‐integrity. We prove several results about edge‐tenacious graphs as well as find numerous classes of edge