Solid Brick: Graph Theory & Finite Element

• Solid Brick is a dual-use term defining both a class of graphs with specific perfect matching constraints and a three-dimensional finite element used in structural modeling.
• In graph theory, a solid brick is a matching-covered graph where the removal of two vertex-disjoint odd cycles results in no perfect matching, emphasizing its unique connectivity properties.
• In finite element analysis, the eight-node solid brick uses trilinear Lagrange interpolation and 24 displacement DOF to accurately simulate solid bodies without geometric simplification.

Searching arXiv for recent and relevant papers on both graph-theoretic and finite-element uses of “solid brick.” Solid brick denotes two distinct technical objects in current mathematical and engineering usage. In graph theory, a solid brick is a brick $G$ such that, whenever $C_1$ and $C_2$ are vertex-disjoint odd cycles of $G$, the graph $G-(V(C_1)\cup V(C_2))$ has no perfect matching (Zhang et al., 29 Jul 2025). In finite-element analysis, a solid brick element is a three-dimensional finite element that can model solid bodies and structures without any a priori geometric simplification; the eight-node solid brick element is a standard hexahedral instance with trilinear interpolation and $24$ displacement DOF (Zhang et al., 2023). The term therefore has precise but unrelated meanings in matching theory and computational mechanics.

1. Solid bricks in matching-covered graph theory

A graph $G$ of order $n=|V(G)|$ is matching-covered if it is connected, $n\ge 2$, and every edge of $G$ lies in some perfect matching (Zhang et al., 29 Jul 2025). A nonbipartite matching-covered graph $C_1$0 is a brick if it has no nontrivial tight cut. Equivalently, by Edmonds–Lovász–Pulleyblank (1982),

$C_1$1

A brick $C_1$2 is solid if whenever $C_1$3 are vertex-disjoint odd cycles, then

$C_1$4

This condition isolates a subclass of bricks in which the coexistence of two disjoint odd cycles is tightly constrained by perfect-matching structure (Zhang et al., 29 Jul 2025).

The graph-theoretic notion is intrinsically matching-theoretic rather than geometric. Its defining properties are expressed through connectivity, tight-cut structure, and the existence or nonexistence of perfect matchings after specified deletions. In this sense, solidity is a global obstruction condition imposed on a brick.

2. Removable, $C_1$5-invariant, and solitary edges

In a matching-covered graph $C_1$6, an edge $C_1$7 is removable if $C_1$8 remains matching-covered (Zhang et al., 29 Jul 2025). If $C_1$9 denotes the number of bricks occurring in any tight-cut decomposition of $C_2$0, then a removable edge $C_2$1 is $C_2$2-invariant if

$C_2$3

For a brick $C_2$4, one has $C_2$5, so $C_2$6-invariance means that $C_2$7 is still a single brick.

An edge $C_2$8 is solitary if it belongs to exactly one perfect matching of $C_2$9; otherwise it is nonsolitary. The paper centers on the interaction between $G$0-invariance and solitariness, a relationship posed in a problem of Lucchesi and Murty (Zhang et al., 29 Jul 2025).

Two structural lemmas are central. First, in a solid brick, every removable edge $G$1 satisfies $G$2. Second, in a solid brick on $G$3 vertices, every vertex is incident with at most two nonremovable edges. Combined with the previous lemma, this gives at most two nonsolitary edges at each vertex. These facts sharply restrict local edge behavior and serve as the entry point for the global classification proved in the same work.

3. Characterization by wheels

The main theorem states: let $G$4 be a solid brick of even order $G$5. Then every $G$6-invariant edge of $G$7 is solitary if and only if $G$8 is the wheel $G$9 (Zhang et al., 29 Jul 2025). Here $G-(V(C_1)\cup V(C_2))$0 is obtained by taking an $G-(V(C_1)\cup V(C_2))$1-cycle, called the rim, and joining a new vertex, called the hub, to all rim-vertices by spokes.

The sufficiency direction is explicit. If $G-(V(C_1)\cup V(C_2))$2 with $G-(V(C_1)\cup V(C_2))$3 even, then the only removable edges are the $G-(V(C_1)\cup V(C_2))$4 spokes, and each such spoke $G-(V(C_1)\cup V(C_2))$5 is $G-(V(C_1)\cup V(C_2))$6-invariant. Deleting its ends leaves an odd path, which has exactly one perfect matching; hence each spoke is solitary. Therefore, in an even wheel, every $G-(V(C_1)\cup V(C_2))$7-invariant edge is solitary (Zhang et al., 29 Jul 2025).

The necessity direction excludes other solid bricks. If $G-(V(C_1)\cup V(C_2))$8 is a solid brick of even order $G-(V(C_1)\cup V(C_2))$9 and every $24$0-invariant edge is solitary, then $24$1 cannot be cubic; otherwise Theorem 1.2 of Zhang–Lu–Zhang applies, but those extremal cubic bricks are nonsolid. Hence there is a vertex $24$2 of degree $24$3. By the solid-brick machinery, at most two edges at $24$4 can be nonsolitary, so at least two edges $24$5 and $24$6 are solitary. This yields unique perfect matchings $24$7 and $24$8 of $24$9 and $G$0, respectively, and their symmetric difference is an $G$1–$G$2 alternating path

$G$3

of odd length at least $G$4 (Zhang et al., 29 Jul 2025).

4. Alternating-path rigidity and excluded configurations

The proof then examines how the remainder of the graph can attach to the alternating path $G$5 (Zhang et al., 29 Jul 2025). A sequence of “no-alternating-cycle” lemmas establishes that neither $G$6 nor $G$7 can contain an $G$8-alternating cycle, because the unique-matching property would be contradicted. Likewise, any hypothetical $G$9-alternating path joining two vertices of $n=|V(G)|$0 outside the edges of $n=|V(G)|$1 would create two vertex-disjoint odd cycles whose removal preserves a perfect matching, contradicting solidity.

From these restrictions, every vertex off $n=|V(G)|$2 can attach to $n=|V(G)|$3 only in a highly constrained manner. The paper reduces the possibilities to three candidate configurations $n=|V(G)|$4. Solidness eliminates two of them, and the remaining configuration forces that every rim-vertex of $n=|V(G)|$5 has exactly three neighbors: its two neighbors on $n=|V(G)|$6 and the hub $n=|V(G)|$7. No other vertices exist, so $n=|V(G)|$8 is exactly the wheel with hub $n=|V(G)|$9 and rim $n\ge 2$0 (Zhang et al., 29 Jul 2025).

The same paper records representative examples and exceptions. Wheels $n\ge 2$1 with even $n\ge 2$2 are solid bricks, and every spoke is removable, hence $n\ge 2$3-invariant, and lies in exactly one perfect matching. By contrast, the exceptional bricks $n\ge 2$4, $n\ge 2$5, and the Petersen graph are not wheels, and in those some $n\ge 2$6-invariant edges fail to be solitary or the graph is small. The cubic extremal bricks $n\ge 2$7 of Zhang–Lu–Zhang are non-solid, because in each such graph one finds two disjoint odd cycles whose removal leaves a perfect matching. This resolves the solid-brick case of the Lucchesi–Murty problem without conflict from the cubic extremal family.

5. The eight-node solid brick element

In computational mechanics, the solid brick element is defined as a three-dimensional finite element that can model solid bodies and structures without any a priori geometric simplification (Zhang et al., 2023). The eight-node solid brick, or hexahedron, occupies a trilinear quadrilateral in the reference $n\ge 2$8 space with corners at $n\ge 2$9.

Quantity	Value
Number of nodes $G$0	8
Degrees of freedom per node $G$1	3
Total element DOF $G$2	24

The standard node numbering and local coordinates are: $G$3

$G$4

Its trilinear Lagrange shape functions are

$G$5

where $G$6 are the nodal coordinates. Explicitly, for the four bottom nodes,

$G$7

$G$8

and the top four $G$9 are obtained by replacing $C_1$00 in the above (Zhang et al., 2023).

6. Assumed-stress formulation, stiffness decomposition, and quadrature

The stiffness construction is developed from an assumed Stress Method whose formulation is based on the Hellinger–Reissner principle developed according to Kang’s study in 1986 (Zhang et al., 2023). The two-field Hellinger–Reissner functional is

$C_1$01

The stress interpolation is assumed in the form

$C_1$02

where $C_1$03 is a $C_1$04 polynomial geometric matrix.

Traction on each face yields nodal forces linearly in $C_1$05,

$C_1$06

with

$C_1$07

The explicit symbolic $C_1$08 matrix is given in equation (7) of the paper. The complementary energy in $C_1$09 is

$C_1$10

with flexibility

$C_1$11

Its inverse $C_1$12 is the generalized stress-parameter stiffness, $C_1$13. Stationarity with respect to $C_1$14 yields

$C_1$15

so the element’s consistent stiffness is

$C_1$16

Following Felippa (2013), the consistent stiffness can be algebraically split as

$C_1$17

with

$C_1$18

and

$C_1$19

where $C_1$20 is the physical volume (Zhang et al., 2023). The basic stiffness $C_1$21 has rank $C_1$22 and exactly reproduces constant-strain (mix-ability) and rigid-body modes. The high-order stiffness $C_1$23 has rank $C_1$24; $C_1$25 is orthogonal to the six rigid-body and six constant-strain modes, expressed as $C_1$26.

The mathematical requirements are stated explicitly. For consistency and mix-ability, $C_1$27 reproduces the exact strain energy for any constant-strain $C_1$28, implying

$C_1$29

whenever $C_1$30 is uniform. For stability, $C_1$31, with six zero modes corresponding only to rigid-body translations and rotations. For accuracy, bending tests on all six faces yield unity energy ratios under exact beam-bending displacement patterns; equations (13)–(14) are cited for these checks.

The numerical integration prescription separates the two stiffness contributions. The basic stiffness $C_1$32 is assembled analytically via $C_1$33 and $C_1$34, so no volume quadrature is needed. The high-order part $C_1$35 arises from integrals of up to fourth-order polynomials in $C_1$36. A $C_1$37 Gauss–Legendre rule exactly integrates up to third-degree polynomials, whereas full integration of quartic terms in $C_1$38 uses a $C_1$39 rule with points at $C_1$40 and weights $C_1$41 (Zhang et al., 2023). The recommended scheme is therefore $C_1$42 exact by construction and $C_1$43 integrated by $C_1$44 for many engineering cases or $C_1$45 for full quartic accuracy.