Global edge alliances in graphs

Abstract

In the paper we introduce and study a new problem of finding a minimum global edge alliance in a graph which is related to the global defensive alliance (Haynes et al., 2013; Hedetniemi, 2004) and the global defensive set (Lewoń et al., 2016). We proved the NP-completeness of the global edge alliance problem for subcubic graphs and we constructed polynomial time algorithms for trees. We found the exact values of the size of the minimum global edge alliance for certain classes: paths, cycles, wheels, complete k-partite graphs and complete k-ary trees. Moreover, we proved some lower bounds for arbitrary graphs.