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arXiv · 2611.09072

Claw-free minimal factor-critical and bicritical graphs

Huaibin Zhou·Mingyang Guo·Xiumei Wang
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Abstract

A graph $G$ is called $k$-factor-critical if $G-S$ has a perfect matching for every subset $S\subseteq V(G)$ with $|S|=k$, and is minimal if $G-e$ is not $k$-factor-critical for every edge $e$ of $G$. The cases $k=1$ and $k=2$ correspond to factor-critical and bicritical graphs, respectively. In this paper, we obtain complete structural characterizations of claw-free minimal factor-critical graphs and claw-free minimal bicritical graphs. The former is characterized by ear decompositions. For the latter, we first characterize the $3$-connected case using the theory of line graphs and then extend the characterization to the general case by a suitable $2$-sum operation.

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Huaibin Zhou, Mingyang Guo, Xiumei Wang. 2026-11-06. Claw-free minimal factor-critical and bicritical graphs. https://arxiv.org/abs/2611.09072

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