Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On essentially 4-edge-connected cubic bricks   Lov\'asz (1987) proved that every matching covered graph $G$ may be uniquely
decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let
$b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also
matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is
quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is
quasi-$b$-invariant.)
  A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that
$G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable
doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three
removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture
of Lov\'asz which states that every brick, distinct from $K_4$,
$\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge.
  A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected
and if its only $3$-cuts are the trivial ones; it is well-known that each such
graph is either a brick or a brace; we provide a graph-theoretical proof of
this fact.
  We prove that if $G$ is any essentially $4$-edge-connected cubic brick then
its edge-set may be partitioned into three (possibly empty) sets: (i) edges
that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii)
quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent
quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen
graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$
(distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce
that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$,
distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.

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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat