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.

• The paper completely characterizes near-bipartite bricks with the forcing-edge property as five families, G₁*, G₃*, G₈*, G₉*, and G₁₀*, excluding K₄, the complement of C₆, and the Petersen graph.
• It proves that, for near-bipartite bricks, every b-invariant edge is forcing if and only if the graph is extremal, meaning its perfect matchings form a basis of the matching lattice.
• The structural analysis shows that any such brick has at most 12 vertices, while allowing precisely the specified parallel-edge extensions of the five small base graphs.

Background and problem statement

A connected graph with at least one edge in which every edge lies in some perfect matching is matching covered. Lovász's tight cut decomposition expresses every matching covered graph uniquely, up to multiple edges, as a list of bricks (nonbipartite members) and braces; the number of bricks $b(G)$ is the brick number. An edge $e$ is removable if $G-e$ remains matching covered, and $b$-invariant if it is removable and $b(G-e)=b(G)$. A forcing edge (or solitary edge) lies in exactly one perfect matching. De Carvalho, Lucchesi, and Murty showed that every brick other than $K_4$, $\overline{C_6}$, and the Petersen graph has a $b$-invariant edge, and that extremal matching covered graphs—those whose perfect matchings form a basis of the matching lattice—are characterized by the property that every $b$-invariant edge is solitary (2607.00608).

Motivated by this connection, Lucchesi and Murty posed the problem of characterizing all bricks, excluding the three exceptional graphs above, in which every $b$-invariant edge is a forcing edge. Prior partial results cover extremal bricks (where the property holds by definition), simple solid bricks (only odd wheels), claw-free bricks (exactly four graphs), and cubic bricks (seven graphs). This paper resolves the problem for the class of near-bipartite bricks: nonbipartite matching covered graphs containing a removable doubleton $e$0 whose deletion yields a bipartite matching covered graph.

Main result

The central theorem gives a complete characterization. Let $e$1 denote the family of graphs obtained from the underlying simple graph $e$2 by allowing arbitrary multiplicities on the dashed edge classes depicted in the paper's figure, for $e$3.

Theorem. Let $e$4 be a near-bipartite brick distinct from $e$5, $e$6, and the Petersen graph. Every $e$7-invariant edge of $e$8 is a forcing edge if and only if $e$9 for some $G-e$0.

The five base graphs are small: $G-e$1 has 8 vertices and 12 edges, $G-e$2 has 10 vertices and 15 edges, $G-e$3 has 6 vertices and 10 edges, $G-e$4 has 8 vertices and 13 edges, and $G-e$5 has 10 vertices and 16 edges. Two of them ($G-e$6, $G-e$7) are cubic and were already covered by the earlier cubic-brick classification; the remaining three are genuinely new to this theorem.

The proof of sufficiency exploits the fact that the union of perfect matchings containing forcing edges equals the full edge set of each $G-e$8: any removable edge outside that union would have to be $G-e$9-invariant yet non-forcing, which the covering argument rules out. Multiple-edge extensions preserve the property because each copy of a dashed-class edge is itself forcing.

Structural consequences

Two corollaries sharpen the picture considerably:

1. Extremality equivalence. Every graph in each $b$0 is an extremal brick: direct computation confirms $b$1 for all five base graphs, and replacing a dashed edge by $b$2 parallel edges increases both sides by $b$3. Combined with the main theorem, this yields: a near-bipartite brick is extremal if and only if all its $b$4-invariant edges are forcing. This is a strong statement—for this class, Lucchesi and Murty's property collapses exactly to extremality, leaving no intermediate cases.
2. Bounded size. Any near-bipartite brick satisfying the property has at most 12 vertices. The proof establishes the structural identity $b$5 and $b$6 for the bipartition $b$7 of $b$8, bounding each part by 6 vertices. Consequently, the characterization is finite and effectively verifiable.

The key structural lemma driving these conclusions states that if $b$9 is a forcing edge of $b(G-e)=b(G)$0 (with $b(G-e)=b(G)$1, $b(G-e)=b(G)$2), then $b(G-e)=b(G)$3 has a unique perfect matching, hence contains degree-one vertices in both parts by the classical bipartite unique-matching theorem; 3-connectivity then forces those vertices to have degree exactly 3 in $b(G-e)=b(G)$4, placing them in the sets $b(G-e)=b(G)$5 and $b(G-e)=b(G)$6. Notably, neither edge of the removable doubleton $b(G-e)=b(G)$7 can be forcing, since each lies in a perfect matching of $b(G-e)=b(G)$8 distinct from any fixed one.

Method of the necessity proof

The necessity direction proceeds by case analysis on vertex degrees within $b(G-e)=b(G)$9. A first claim shows every vertex of $K_4$0 has degree at most 4: if some $K_4$1 had degree $K_4$2, two incident $K_4$3-invariant forcing edges would force two triangles through $K_4$4, pinning the structure to a specific graph $K_4$5 in which certain vertices are incident with no forcing edge—contradicting Lemma on non-$K_4$6-invariant incidence. For non-cubic simple $K_4$7, three degree patterns are analyzed:

• Pattern (i) ($K_4$8, $K_4$9): forces $\overline{C_6}$0 and $\overline{C_6}$1.
• Pattern (ii) ($\overline{C_6}$2, $\overline{C_6}$3): forces $\overline{C_6}$4; two candidate graphs $\overline{C_6}$5 are eliminated because a vertex $\overline{C_6}$6 carries a $\overline{C_6}$7-invariant edge but no forcing edge, leaving only $\overline{C_6}$8.
• Pattern (iii) (all four vertices of degree 3): subclaims progressively exclude $\overline{C_6}$9 and $b$0 via alternating-cycle arguments showing candidate $b$1-invariant edges lie on alternating cycles and hence cannot be forcing; the case $b$2 further requires $b$3 and the existence of two disjoint triangles, ultimately yielding $b$4.

For graphs with multiple edges, the property transfers to the underlying simple graph, and multiplicity analysis shows parallel edges may be added only to the designated dashed classes while retaining near-bipartiteness (at least one removable doubleton must remain single).

Limitations and open questions

The result is confined to near-bipartite bricks; the general Problem of Lucchesi and Murty remains open for arbitrary bricks, and the paper does not address it beyond noting the existing partial results for solid, claw-free, and cubic bricks. The classification depends on the previously established bound of $b$5 $b$6-invariant edges for near-bipartite bricks with at least six vertices, so any refinement of that counting result could potentially simplify or extend the present argument. Whether an analogous finite classification exists for bricks that are neither near-bipartite nor in the already-settled classes is left unresolved.

Conclusion

This paper settles the Lucchesi–Murty problem for near-bipartite bricks, identifying exactly five families $b$7 ($b$8) of small graphs—all of them extremal—as the complete list. The equivalence between "every $b$9-invariant edge is forcing" and extremality within this class, together with the 12-vertex size bound, indicates that the forcing property is highly restrictive for near-bipartite bricks. The remaining challenge is extending the characterization to general bricks not covered by existing partial classifications.