On the anti-Kelul\'{e} problem of cubic graphs

An edge set $S$ of a connected graph $G$ is called an anti-Kekul'e set if $G-S$ is connected and has no perfect matchings, where $G-S$ denotes the subgraph obtained by deleting all edges in $S$ from $G$. The anti-Kekul'e number of a graph $G$, denoted by $ak(G)$, is the cardinality of a smallest anti-Kekul'e set of $G$. It is NP-complete to find the smallest anti-Kekul'e set of a graph. In this paper, we show that the anti-Kekul'{e} number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekul'{e} number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to find all smallest anti-Kekul'{e} sets of a connected cubic graph.