Nice vertices in cubic graphs

A subgraph $G^{\prime }$ of a graph G is nice if $G-V\left(G^{\prime }\right)$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex u of a cubic graph is nice if u and its neighbors induce a nice subgraph. D. Král et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph G is 2-connected, then G has at least 4 nice vertices; if G is 3-connected and $G\ne K_4$, then G has at least 6 nice vertices. We also determine all the corresponding extremal graphs. For a cubic bipartite graph G with bipartition $(A,B)$, a pair of vertices $a\in A$ and $b\in B$ is called a nice pair if a and b together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph G is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that G has at least 9 nice pairs of vertices and $K_{3,3}$ is the only extremal graph.