Note on fair game edge-connectivity of graphs.

Lately, Matsumoto and Nakamigawa introduced a game invariant concerning the edge-connectivity of graphs, called the game edge-connectivity (Matsumoto and Nakamigawa, 2021). To define the invariant, they introduced a combinatorial game where the first player deletes one edge and the second one contracts (or protects) one edge. In this paper, considering a natural variation of the game in which the second player also deletes an element instead of protecting one, we introduce an alternative game invariant, called the fair game edge-connectivity of graphs. By showing that the fair game edge-connectivity of every 2-edge-connected graph G is equal to 2 & lambda;(G)-1 except some graphs, where & lambda;(G) denotes the edge-connectivity of G, we could completely determine the fair game edge-connectivity of all graphs.& COPY; 2023 Elsevier B.V. All rights reserved.