Tight cuts in matching covered graphs

Abstract: An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and Pulleyblank [Brick decompositions and the matching rank of graphs, Combinatorica 2(3) (1982) 247-274] proved that if a matching covered graph has a non-trivial tight cut, then it also has a non-trivial ELP-cut. In confirmation of a conjecture proposed by Carvalho, Lucchesi and Murty, Chen et. al. [Laminar tight cuts in matching covered graphs, J. Comb. Theory, Ser. B, 150 (2021) 177-194] showed that if C is a non-trivial tight cut of a matching covered graph G, then G has at least one C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C. In this paper, we present a complete characterization of non-trivial tight cuts in matching covered graphs, from which the result of Chen et. al. can be derived directly. Moreover, we show that the lower bound of the number of C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C in the result of Chen et. al. is sharp.

• The paper proves that every non-trivial tight cut in a matching covered graph is either a barrier-cut or an essential generalized 2-separation (GS) cut, extending earlier results beyond graphs with at most two bricks.
• The paper establishes that every essential GS-cut is tight and that its structure guarantees at least one laminar ELP-cut, yielding Chen et al.’s theorem as a direct corollary.
• The paper shows the lower bound is sharp: GS-cuts can have exactly two laminar ELP-cuts, while essential GS-cuts that are not GS-cuts can have exactly one.

Background and motivation

A connected non-trivial graph $G$ is matching covered if every edge lies in some perfect matching. An edge cut $\partial(X)$ is tight if every perfect matching intersects it in exactly one edge. Tight cuts are the engine of Lovász's tight cut decomposition, which reduces any matching covered graph to braces (bipartite) and bricks (non-bipartite); the number of resulting bricks is an invariant of $G$. The two canonical families of tight cuts are the ELP-cuts: barrier-cuts $\partial(V(Q))$, where $B$ is a barrier ($o(G-B)=|B|$) and $Q$ is an odd component of $G-B$, and 2-separation cuts associated with a 2-separation $\{u,v\}$ whose removal leaves only even components. The ELP-Theorem of Edmonds, Lovász and Pulleyblank states that every matching covered graph with a non-trivial tight cut has a non-trivial ELP-cut [ELP82].

The paper under review addresses the finer question of how an arbitrary non-trivial tight cut relates to ELP-cuts. Carvalho, Lucchesi and Murty conjectured that every non-ELP tight cut admits a laminar non-trivial ELP-cut; Chen et al. confirmed this by showing that for any non-trivial tight cut $C$, the family $\partial(X)$0 of $\partial(X)$1-sheltered barrier-cuts and laminar 2-separation cuts satisfies $\partial(X)$2 [CFLLZ20]. That result was previously established only for bicritical graphs and graphs with brick number at most two [CLM18]. The present paper strengthens this line of work in two ways: it gives a complete structural characterization of all non-trivial tight cuts, from which Chen et al.'s theorem follows directly, and it proves that the lower bound of one is sharp.

Essential GS-cuts

The central new notion is the generalized 2-separation cut (GS-cut). For a cut $\partial(X)$3, let $\partial(X)$4 be the set of 2-separations $\partial(X)$5 with $\partial(X)$6. Then $\partial(X)$7 is a GS-cut if: (1) there exists a chain structure among members of $\partial(X)$8 (the statement in the source text is garbled here, but the intent is a connectivity condition on the associated 2-separations), and (2) whenever two such 2-separations $\partial(X)$9 intersect in exactly one vertex with nested component sets $G$0, all vertices of each odd component of $G$1 lie on the shore opposite to the shared vertex, and all vertices of each even component lie on a common shore. A 2-separation cut is the special case $G$2.

An essential GS-cut relaxes this: after contracting a set $G$3 of pairwise disjoint $G$4-sheltered non-trivial barriers into single vertices $G$5, the image $G$6 must be a GS-cut in which every contracted vertex $G$7 lies in some 2-separation meeting $G$8 in exactly one vertex. This generalizes the framework of Carvalho–Lucchesi–Murty, who had shown that in brick number at most two, every non-trivial tight cut is either a barrier-cut or an essential GS-cut [CLM17].

Two technical results support the characterization. First, Proposition (tightness across splicing): if $G$9 arises by edge-splicing two subgraphs along a 2-separation $\partial(V(Q))$0 with $\partial(V(Q))$1, then $\partial(V(Q))$2 is tight in $\partial(V(Q))$3 exactly when both $\partial(V(Q))$4 are tight in their respective pieces. Second, Proposition: every essential GS-cut is tight. The proof first shows by induction on the number $\partial(V(Q))$5 of crossing 2-separation cuts that every GS-cut is tight — peeling off end-2-separations and applying the splicing proposition — then lifts tightness through barrier contractions using the standard fact that tight cuts persist under contraction.

Sharpness of the laminar ELP bound

The paper proves that any non-trivial GS-cut $\partial(V(Q))$6 of a matching covered graph satisfies $\partial(V(Q))$7: choosing an end-2-separation $\partial(V(Q))$8 of the associated family yields one laminar 2-separation cut, and either its complement also yields one, or a second end-2-separation $\partial(V(Q))$9 does. This bound is sharp. The graph $B$0, built from two paths $B$1 and $B$2 with chords $B$3 plus rungs $B$4, is bicritical (an edge-splice of $B$5 copies of $B$6), so all its barriers are trivial and its only ELP-cuts are 2-separation cuts. For the GS-cut $B$7, exactly two ELP-cuts are laminar with it. Moreover, a variant $B$8 shows that even for essential GS-cuts that are not GS-cuts, $B$9 can occur when a single sheltered barrier contracts to a vertex lying in every relevant 2-separation. Hence no uniform improvement of Chen et al.'s lower bound is possible.

Main theorem and proof outline

The main result is:

Theorem. In a matching covered graph, every non-trivial tight cut is either a barrier-cut or an essential GS-cut.

This extends the CLM dichotomy from brick number at most two to all matching covered graphs, and constitutes a complete characterization of non-trivial tight cuts. The proof proceeds by induction on the number of non-trivial tight cuts in $o(G-B)=|B|$0. If $o(G-B)=|B|$1 has exactly one, the ELP-Theorem forces it to be a barrier-cut (a single 2-separation produces at least two 2-separation cuts). Otherwise, assuming $o(G-B)=|B|$2 is not a barrier-cut, the argument establishes via a sequence of claims that $o(G-B)=|B|$3 contains a $o(G-B)=|B|$4-sheltered non-trivial barrier or a 2-separation cut laminar with $o(G-B)=|B|$5:

• In the bicritical case, a minimal 2-separation $o(G-B)=|B|$6 with an even component $o(G-B)=|B|$7 is shown to cross $o(G-B)=|B|$8 in a controlled way; contracting the opposite side yields a smaller graph where the ELP-Theorem produces an ELP-cut that lifts back to a contradiction.
• When $o(G-B)=|B|$9 has a maximal non-trivial barrier $Q$0 with components $Q$1 of $Q$2 (odd intersections with $Q$3 for $Q$4), counting arguments against perfect matchings show $Q$5, $Q$6, then $Q$7, so $Q$8; this forces some $Q$9 to be a non-trivial tight cut of a contraction that is not a barrier-cut, hence by induction an essential GS-cut, whose structure ultimately yields a laminar 2-separation cut in $G-B$0 — contradicting the assumption.

With Claim 1 established, Case 1 (a $G-B$1-sheltered barrier exists) shows $G-B$2 remains an essential GS-cut after contracting the complement of a suitable component of $G-B$3; Case 2 (a laminar 2-separation cut $G-B$4 exists) splits according to whether the contracted cut $G-B$5 is a barrier-cut — in which case a constructed barrier $G-B$6 exhibits $G-B$7 as an essential GS-cut — or not, in which case induction applies and the 2-separations of $G-B$8 containing $G-B$9 or $\{u,v\}$0 are analyzed case by case, including the situation where $\{u,v\}$1 is an essential but not plain GS-cut with associated barriers $\{u,v\}$2.

Notably, the proof does not depend on Chen et al.'s theorem; rather, since a barrier-cut is trivially an ELP-cut laminar with itself, and an essential GS-cut always carries at least one laminar ELP-cut, Theorem (lam) follows as a corollary of the main theorem.

Limitations and open questions

The characterization is existential and structural rather than algorithmic: the paper does not address the complexity of recognizing GS-cuts or essential GS-cuts, nor of finding the associated barriers and 2-separation families. The definition of a GS-cut involves conditions on nested pairs of 2-separations whose precise formulation requires care (the source text's condition (1) appears incomplete as typeset). The sharpness examples show the count $\{u,v\}$3 cannot be uniformly bounded above one, but the exact minimum as a function of the number of associated barriers — e.g., whether $\{u,v\}$4 in general — is left unexamined. Finally, the paper does not explore consequences for the matching lattice or for removable ear decompositions, areas where tight cut structure typically plays a role.

Conclusion

This paper settles the structure of non-trivial tight cuts in full generality: every such cut in a matching covered graph is either a barrier-cut or an essential generalized 2-separation cut, extending a dichotomy previously known only for graphs with at most two bricks. The characterization yields Chen et al.'s laminar ELP-cut theorem as a direct corollary, independently of its original proof, and demonstrates via explicit bicritical constructions that the guaranteed number of laminar ELP-cuts is exactly one in the worst case. The result consolidates the structural theory underlying tight cut decomposition and provides a sharper vocabulary — GS-cuts and their barrier-contracted relatives — for future work on bricks and braces.