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Bricks in which every vertex is incident with a forcing edge
Abstract
A matching covered graph is a brick if it is 3-connected and bicritical. An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. We prove that every vertex of a simple brick G is incident with a forcing edge if and only if G is an odd wheel or is isomorphic to one of the following three graphs: the triangular prism \overline{C_6},\overline{C_6}^+ or the bicorn H_8. Here \overline{C_6}^+ is obtained from \overline{C_6} by adding an edge between two nonadjacent vertices.
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BibTeX
@misc{dai2026bricks,
title = {{Bricks in which every vertex is incident with a forcing edge}},
author = {Xinyu Dai and Fuliang Lu and Yaxian Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.26594},
eprint = {2606.26594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26594},
}
RIS
TY - UNPB
TI - Bricks in which every vertex is incident with a forcing edge
AU - Xinyu Dai
AU - Fuliang Lu
AU - Yaxian Zhang
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2606.26594
UR - https://arxiv.org/abs/2606.26594
AB - A matching covered graph is a brick if it is 3-connected and bicritical. An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. We prove that every vertex of a simple brick G is incident with a forcing edge if and only if G is an odd wheel or is isomorphic to one of the following three graphs: the triangular prism \overline{C_6},\overline{C_6}^+ or the bicorn H_8. Here \overline{C_6}^+ is obtained from \overline{C_6} by adding an edge between two nonadjacent vertices.
ER -