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The graph parameter neighbor-connectivity was first investigated by Gunther and Hartnell in 1987 and provides important information on how reliable a network can be when failures of a node may impact its neighbors. In this model, the failure of a node causes the deletion of its closed neighborhood, i.e., the node and its adjacent neighbors as well. The minimum number of closed neighborhoods whose removal results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Since then component order connectivity models have emerged to address inadequacies in the traditional models of connectivity, namely, that in many real-world networks, when disconnecting a graph one may be left with components that are large enough to still be operable. By adapting neighbor-connectivity to a component order model, we introduced neighbor-component order connectivity, defined as the minimum number of closed neighborhoods that must be removed from a network to ensure all remaining components have order less than some given threshold value. Given a threshold value of one, the neighbor-component order connectivity of a graph is equivalent to the well-known parameter domination number of the graph. The problem of computing the domination number of an arbitrary graph is NP-hard, and computing neighbor-component order connectivity of a graph for an arbitrary threshold value is also NP-hard. Here, we present a linear-time algorithm for computing the neighbor-component order connectivity of an arbitrary tree for an arbitrary threshold value, thus generalizing the classic linear algorithm of Cockayne, Goodman, and Hedetniemi for the domination number of the tree.

Dr. Charles L. Suffel (1941Suffel ( -2021 was an influential mathematics educator and scholar at Stevens Institute of Technology for more than half a century. Managing Editor of Networks for 20 years, Suffel's reach extended far beyond the Stevens campus. He coauthored dozens of graph theory papers and mentored more than a dozen Ph.D. thesis students. In this article, we discuss his contributions to the field of network reliability theory and his legacy as a teacher and mentor. KEYWORDSAll-terminal reliability, component order connectivity, component order edge connectivity, component order reliability, compression, neighbor-component order connectivity, spanning trees, weighted component edge connectivity SUFFEL'S RESEARCHCharlie Suffel received his Ph.D. from Brooklyn Polytechnical University in 1969 under the direction of the functional analyst George Bachmann. He also worked for a stint at Bell Labs before joining Stevens Institute of Technology as an assistant professor. The first half dozen publications of his academic career were in the area of functional analysis and topological vector spaces. Promoted to full professor in 1979, Suffel would go on to co-author more than more than 50 papers in the areas of graph theory and network reliability theory. The first of these studied subgraphs of Eulerian graphs [14]; coauthored with Boesch and Ralph Tindell, it appeared in the first edition of Journal of Graph Theory. In fact, his most frequent collaborators were Boesch (27 co-authored papers), Dan Gross (23), John T. Saccoman (13) and Bill Kazmierczak (12), the latter two being among Suffel's 15 completed Ph.D. students. What should be noted is that Suffel continued to be an active researcher during his 20 years as the Dean of Graduate Studies at Stevens; in fact, 30 of his papers were published during that time, and more