AI 中文总结
本文研究满足所有可移除边均为孤立边的砖图，证明其非实心版本可通过反复拼接奇轮得到，且无法用K₄拼接替代。
AI 中文摘要
砖图（brick）是指满足对任意两个不同顶点u,v∈V(G)，G-u-v都存在完美匹配的3-连通图G。匹配覆盖图G中的边e若满足G-e仍为匹配覆盖图，则称e是可移除边。砖图G中的可移除边e若满足b(G-e)=b(G)=1，则称e是b-不变边，其中b(H)表示匹配覆盖图H的紧割分解中砖图的数量。图中的孤立边是指仅属于一个完美匹配的边。Lucchesi与Murty提出问题：刻画除K₄、$\boldsymbol{\bar{C}_6}$和Petersen图之外，所有b-不变边均为孤立边的砖图。注意所有b-不变边都是可移除边。本文通过要求所有可移除边均为孤立边来强化该条件，证明满足此强化条件的每个非实心砖图都可通过反复拼接奇轮（允许多重边）得到；此外，这类砖图的性质表明，“反复拼接奇轮”无法替换为“反复拼接K₄副本”。
英文摘要
A brick is a 3-connected graph $G$ such that $G-u-v$ has a perfect matching for any two distinct vertices $u,v\in V(G)$. An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is matching covered. We say that a removable edge $e$ in a brick $G$ is $b$-invariant if $b(G-e)=b(G)=1$, where $b(H)$ denotes the number of bricks in the tight cut decomposition of a matching covered graph $H$. An edge of a graph is solitary if it lies in precisely one perfect matching. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. Note that every $b$-invariant edge is removable. In this paper, we strengthen the condition by requiring that every removable edge is solitary. We show that every nonsolid brick satisfying this strengthened condition can be obtained by repeatedly splicing odd wheels (up to multiple edges). Moreover, properties of such bricks imply that "repeatedly splicing odd wheels" cannot be replaced by "repeatedly splicing copies of $K_4$".
