A note on removable edges in near-bricks

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.


Introduction
All the graphs considered in this paper may have multiple edges, but no loops.We follow Bondy and Murty (2008) for undefined notations and terminologies.Let G be a graph with the vertex set V (G) and the edge set E(G).We denote by ∆(G), or simply ∆, the maximum degree of the vertices of G.For a subset X of V (G), let G[X] denote the subgraph of G induced by X.A matching of a graph is a set of pairwise nonadjacent edges.A perfect matching is one which covers every vertex of the graph.A nontrivial connected graph is matching covered if each edge lies in a perfect matching of the graph.Clearly, every matching covered graph different from K 2 is 2-connected.
For a nonempty proper subset X of V (G), let ∂(X) denote the set of all the edges of G with one end in X and the other end in X, where X := V (G) \ X.The set ∂(X) is called a cut of G, the sets X and X its shores.The shore X of ∂(X) is bipartite if the induced subgraph G[X] is bipartite.A cut is trivial if one of its shores is a singleton, and is nontrivial otherwise.We denote by G/(X → x), or simply G/X, the graph obtained from G by shrinking X to a single vertex x.Similarly, we denote by G/(X → x), or simply G/X, the graph obtained from G by shrinking X to a single vertex x.The two graphs G/X and G/X are called the two ∂(X)-contractions of G.
Let G be a matching covered graph.A cut C of G is tight if |M ∩ C| = 1 for each perfect matching M of G.A matching covered graph which is free of nontrivial tight cuts is a brace if it is bipartite, and is a brick otherwise.If G has a nontrivial tight cut C, then each C-contraction of G is a matching covered graph that has strictly fewer vertices than G. Continuing in this way, we can obtain a list of 2 Deyu Wu et.al matching covered graphs without nontrivial tight cuts, which are bricks and braces.This procedure is known as a tight cut decomposition of G.In general, a matching covered graph may admit several tight cut decompositions.Lovász (1987) showed that any two tight cut decompositions of a matching covered graph yield the same list of bricks and braces (up to multiple edges).This implies that the number of bricks is uniquely determined by G. Let b(G) denote the number of the bricks of G.Note that b(G) = 0 if and only if G is bipartite.
A graph G is a near-brick if it is a matching covered graph with b(G) = 1.Clearly, a near-brick is 2-connected and a brick is a near-brick.A single ear of a graph is a path of odd length whose internal vertices (if any) all have degree two in this graph.A graph is irreducible if it contains no single ear of length three or more.Edmonds et al. (1982) proved that a graph G is a brick if and only if G is 3-connected and G − x − y has a perfect matching for any two distinct vertices x, y ∈ V (G).Therefore, a brick is irreducible.However, a near-brick is not necessarily irreducible.For instance, subdividing an edge of a graph in Figure 1 by inserting two vertices results in a near-brick, which is not irreducible.An edge e of a matching covered graph G is removable if G − e is also matching covered, and is nonremovable otherwise.Clearly, each multiple edge of a matching covered graph is in fact a removable edge.The notion of removable edge is related to ear decompositions of matching covered graphs introduced by Lovász and Plummer.Lovász (1987) showed that every brick distinct from K 4 and C 6 has a removable edge, where K 4 and C 6 are shown in Figure 1.Carvalho, Lucchesi, and Murty proved the following stronger result.
Theorem 1.1 (Carvalho et al. (1999)) Every brick G different from K 4 and C 6 has at least ∆ − 2 removable edges.
The following theorem is our main result which generalizes the above theorem to irreducible nearbricks.
Theorem 1.2 Every irreducible near-brick G different from K 4 and C 6 has at least ∆ − 2 removable edges.
The paper is organized as follows.In Section 2, we present some basic results.In Section 3, we give a proof of Theorem 1.2.

Preliminaries
Lemma 2.1 (Carvalho et al. (1999)) In a brace on six or more vertices, every edge is removable.
Lemma 2.3 (Zhang et al. (2022)) Let C be a tight cut of a matching covered graph G and e an edge of G. Then e is removable in G if and only if e is removable in each C-contraction of G which contains it.
The following equality reveals an important property of the numbers of bricks of matching covered graphs with respect to tight cuts.
Lemma 2.4 (Carvalho et al. (2002)) Let G be a matching covered graph and C a tight cut of G. Let G 1 and G 2 be the two Using the above lemma, we can easily obtain the following result, also see Carvalho et al. (2002).
Lemma 2.5 (Carvalho et al. (2002)) For any tight cut C of a near-brick G, precisely one of the shores of C is bipartite.
To bisubdivide an edge e of a graph G is to replace e by an odd path with length at least three.The resulting graph is called a bisubdivision of G at the edge e.Let RE(G) denote the set of all the removable edges of G.
Lemma 2.6 Let G be a graph and let H be a bisubdivision of G at an edge e. Suppose that H is a matching covered graph.Then G is a matching covered graph with b(G) = b(H) and RE(H) = RE(G)\{e}.
Proof: Since H is a bisubdivision of G at the edge e, H is obtained from G by replacing e by an odd path P with length at least three.We assert that G is not isomorphic to K 2 .Otherwise, H is an odd path, contradicting the assumption that H is a matching covered graph.Let e = uv and X = V (P )\{v}.Then G is isomorphic to H/X.Since P − v is an even path of H with all internal vertices of degree 2 in H, for each perfect matching M of H, we have |M ∩ ∂ H (X)| = 1.Then ∂ H (X) is a tight cut of H. Since H is a matching covered graph, so does G.Since G is not isomorphic to K 2 , G is 2-connected.So u has at least two neighbours in G.This implies that the underlying simple graph of H/X is an even cycle.So b(H/X) = 0.By Lemma 2.4, we have b Now we proceed to show that RE(H) = RE(G)\{e}.Note that each edge of P is incident with a vertex of degree 2 in H, and hence is nonremovable in H.
Then G has at least ∆ − 2 removable edges.To complete the proof, we now show that ∆ 1 ≥ ∆.Clearly, it is true when u ∈ X.We may assume that u ∈ X.
Then u ∈ B and ∆ 1 ≥ d G1 (x) = d G2 (x) ≥ d G (u) − 1 + 2 = ∆ + 1. Theorem 1.2 holds.□Remark.The condition of Theorem 1.2 that the graph is irreducible is necessary.For instance, the graph in Figure2(a) is a near-brick with maximum degree four but not irreducible, and has exactly one removable edge e.Furthermore, the lower bound of Theorem 1.2 is sharp.The graph shown in Figure2(b) is an irreducible near-brick with maximum degree four and has exactly two removable edges e and f ; the graph R 8 shown in Figure2(c) is a cubic brick with exactly one removable edge h.
and then is a removable edge of H/X.Again by Lemma 2.3, we havef ∈ RE(H).It follows that RE(G)\{e} ⊆ RE(H).Consequently, RE(H) = RE(G)\{e}.Suppose that G is an irreducible near-brick different from K 4 and C 6 , and∆ = ∆(G).Then b(G) = 1 and |V (G)| ≥ 4.Moreover, G is 2-connected and matching covered.So δ(G) ≥ 2. If ∆ < 3,then each vertex of G has degree two.Thus G is an even cycle.This implies that b(G) = 0, a contradiction.Therefore, ∆ ≥ 3. We shall show that G has at least ∆ − 2 removable edges by induction on |V (G)| + |E(G)|.Now we consider the following two cases according to whether G has parallel edges or not.Then each edge of C is a removable edge of G 2 .By Lemma 2.3, we have RE(G 1 ) ⊆ RE(G).Since d G1 (x) ≥ 4 and G is irreducible, G 1 is irreducible and is different from K 4 and C 6 .By the induction hypothesis, G 1 has at least ∆ 1