Near-bipartite bricks in which every b-invariant edge is a forcing edge

Abstract

A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lovász proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges. Denote by b(G) the number of bricks in such a decomposition. An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G). Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G. 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 a forcing edge. In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.

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BibTeX RIS

BibTeX

@misc{zhang2026nearbipartite,
  title = {{Near-bipartite bricks in which every b-invariant edge is a forcing edge}},
  author = {Yaxian Zhang and Fuliang Lu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.00608},
  eprint = {2607.00608},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.00608},
}

RIS

TY  - UNPB
TI  - Near-bipartite bricks in which every b-invariant edge is a forcing edge
AU  - Yaxian Zhang
AU  - Fuliang Lu
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2607.00608
UR  - https://arxiv.org/abs/2607.00608
AB  - A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lovász proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges. Denote by b(G) the number of bricks in such a decomposition. An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G). Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G. 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 a forcing edge. In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.
ER  -

Abstract

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