- The paper proves that every simple nonsolid brick in which each removable edge is solitary is a splicing of an odd wheel, possibly with multiple edges, and a smaller brick with the same property.
- It establishes closure criteria for bricks satisfying R=S, showing that splicing preserves the equality under specified hypotheses while counterexamples demonstrate that those assumptions are necessary.
- The paper constructs an infinite family of such bricks that cannot be obtained by repeatedly splicing copies of K4, proving that odd wheels cannot generally be replaced by K4 in the structural theorem.
Background and motivation
A brick is a 3-connected graph G such that G−u−v has a perfect matching for every pair of distinct vertices u,v; equivalently, bricks are the nonbipartite members of the tight cut decomposition of matching covered graphs (2608.12832). An edge e is removable if G−e remains matching covered, and is solitary (or forcing) if it lies in exactly one perfect matching. A removable edge e of a brick G is b-invariant when b(G−e)=b(G)=1, where b(⋅) counts bricks in the tight cut decomposition.
Lucchesi and Murty posed the problem of characterizing bricks, other than G−u−v0, G−u−v1 and the Petersen graph, in which every G−u−v2-invariant edge is solitary. Prior work resolved special cases: Zhang et al. identified exactly seven cubic bricks with this property, Zhang and Wang characterized several claw-free examples, and Zhou et al. gave a complete characterization of claw-free solid bricks. The paper under discussion strengthens the hypothesis from "G−u−v3-invariant" to "removable." Since every G−u−v4-invariant edge is removable, any result for the strengthened condition applies a fortiori to the original problem, while the class of graphs satisfying it is potentially smaller.
Main structural theorem
The central result concerns nonsolid bricks, i.e., bricks possessing separating cuts that are not tight:
Theorem 1. Every simple nonsolid brick in which every removable edge is solitary is a splicing of an odd wheel (up to multiple edges) and a brick in which every removable edge is solitary.
Here a splicing G−u−v5 identifies the incident edge sets of vertices G−u−v6 and G−u−v7 of equal degree via a bijection G−u−v8. The proof proceeds by invoking a robust-cut lemma guaranteeing a separating cut G−u−v9 whose contractions yield a solid brick u,v0, another brick u,v1, and an intermediate bipartite matching covered graph u,v2. Inheritance properties of solitary edges across separating cuts force every removable edge of u,v3 outside the cut to be solitary, which—by combining the fact that each vertex of a solid brick on at least six vertices is incident with at most two nonremovable edges, with the characterization that simple solid bricks whose vertices are almost all incident with solitary edges are odd wheels—implies that the underlying simple graph of u,v4 is an odd wheel, with multiple edges confined as prescribed (all adjacent if u,v5; spokes only otherwise). A key claim then shows u,v6: otherwise some edge of u,v7 outside u,v8 would be removable but not solitary in u,v9 (using the lemma that edges outside the contracted pair are never solitary in bipartite matching covered graphs), contradicting the hypothesis inherited by e0. Consequently e1 is precisely a splicing of the wheel e2 with e3, and Lemma-level arguments show every removable edge of e4 is also solitary, completing the induction-friendly structure.
The theorem has immediate corollaries for the two-wheel case. If e5 is a splicing of odd wheels e6 and e7 (e8) and every removable edge is solitary, then e9 or G−e0; moreover, when G−e1 and G−e2, the splicing vertex on G−e3 cannot be its hub. Thus among splicings of exactly two odd wheels, one factor must be G−e4.
Bricks with G−e5
Writing G−e6 and G−e7 for the sets of removable and solitary edges, the paper studies bricks with G−e8. Odd wheels G−e9 (e0), possibly with multiple hub edges, satisfy this identity. The main closure result states:
Theorem 2. Let e1, where each e2 is a brick with e3. Then e4 is a brick with e5 if and only if e6 is a brick with e7.
The sufficiency argument shows that a nonremovable solitary edge of e8 would be nonremovable—and hence non-solitary—in one factor, contradicting inheritance of solitary status. This extends to the case where one factor is e9 (possibly with multiple edges) satisfying only G0, using removable doubletons of G1 in place of the G2 assumption. Two counterexamples delineate the boundary of the theorem: there exist bricks G3 with G4 whose splice satisfies G5, and a brick G6 with G7 decomposable as a splice of factors both failing G8. Hence neither hypothesis can be dropped unilaterally.
An infinite family excluding G9 splicings
A natural question is whether "odd wheel" in Theorem 1 can be weakened to "b0" throughout—that is, whether every such brick arises by repeatedly splicing copies of b1. The answer is negative. The authors construct an infinite family b2 built from two terminal gadgets b3 and b4 (b5) joined by a chain of layer gadgets b6 with connector vertices b7 and upper/lower boundary paths. Each b8 is a chain of b9 layers between the terminals.
Two claims establish the theorem for this family. First, letting b(G−e)=b(G)=10 be the set of vertices of degree exceeding five (there are b(G−e)=b(G)=11 of them), the solitary edges are exactly those incident with b(G−e)=b(G)=12 but avoiding b(G−e)=b(G)=13; every perfect matching containing such an edge covers all remaining edges, so all other edges are nonremovable, whence every removable edge is solitary. Second, the triangles of b(G−e)=b(G)=14 are precisely those formed at vertices of b(G−e)=b(G)=15 with adjacent neighbor pairs; contracting any such triangle destroys 3-connectivity (yielding either a 2-vertex cut or a vertex of degree two), so no triangle contraction yields a brick. Since any splicing with b(G−e)=b(G)=16 up to multiple edges would require a triangle whose contraction is a brick, no b(G−e)=b(G)=17 admits such a decomposition. Consequently, the odd wheel in Theorem 1 genuinely cannot be replaced by b(G−e)=b(G)=18.
Limitations and open questions
The paper's results are conditional on simplicity in Theorem 1 (multiple edges enter only via the "up to multiple edges" qualification on wheels), and the structural description is recursive rather than a closed-form classification: the residual factor b(G−e)=b(G)=19 is again a brick in which every removable edge is solitary, so a full classification still requires understanding the solid case (odd wheels, per prior work) together with termination of the splicing process. The family b(⋅)0 settles the b(⋅)1 question negatively but does not characterize which pairs of odd wheels can appear as terminal gadgets or which bijections b(⋅)2 preserve the solitary-removable property. Whether the planar analogue—that every planar brick in which every removable edge is solitary contains a triangle—extends beyond the sketched arguments, and whether the seven cubic bricks and claw-free families combine with Theorem 1 into a complete characterization of the original Lucchesi–Murty problem, remain open.
Conclusion
This paper strengthens the Lucchesi–Murty solitary-edge program from b(⋅)3-invariant to removable edges and proves that simple nonsolid bricks with the strengthened property are exactly iterated splicings of odd wheels with smaller bricks of the same type, with the wheel factor irreducible to b(⋅)4 as witnessed by an explicit infinite family. The closure analysis of the condition b(⋅)5 under splicing, including its sharpness via counterexamples, provides the technical machinery underlying these structural results and frames the remaining classification questions.
