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

TL;DR

This paper investigates the anti-Kekulé problem in cubic graphs, establishing bounds on the anti-Kekulé number and providing a polynomial-time algorithm to find minimal anti-Kekulé sets.

Contribution

It determines the anti-Kekulé number for 2-connected and bipartite cubic graphs and introduces an efficient algorithm for finding minimal anti-Kekulé sets.

Findings

Anti-Kekulé number of 2-connected cubic graphs is 3 or 4.

Anti-Kekulé number of connected cubic bipartite graphs is always 4.

Polynomial-time algorithm for finding all smallest anti-Kekulé sets.

Abstract

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.